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\title{Small Furstenberg sets}

\author{Ursula Molter and Ezequiel Rela}

\address{Departamento de
Matem\'atica \\ Facultad de Ciencias Exactas y Naturales\\
Universidad de Buenos Aires\\ Ciudad Universitaria, Pabell\'on
I\\ 1428 Capital Federal\\ ARGENTINA\\ and IMAS - CONICET, Argentina}
 \email[Ursula Molter]{umolter@dm.uba.ar}
\email[Ezequiel Rela]{erela@dm.uba.ar} \keywords{Furstenberg sets, Hausdorff
dimension, dimension function, Jarn\'ik's theorems.} \subjclass{Primary 28A78, 28A80}

\thanks{This research  is partially supported by
Grants: PICT2006-00177, UBACyT X149 and CONICET PIP368}

\begin{abstract}
For $\alpha$ in $(0,1]$, a subset $E$ of $\RR$ is called
\textit{Furstenberg set} of type $\alpha$ or $F_\alpha$-set if
for each direction $e$ in the unit circle there is a line segment
$\ell_e$ in the direction of $e$ such that the Hausdorff
dimension of the set $E\cap\ell_e$ is greater or equal than
$\alpha$. In this paper we show that if $\alpha > 0$,
there exists a set $E\in F_\alpha$ such that $\HH{g}(E)=0$ for
$g(x)=x^{\frac{1}{2}+\frac{3}{2}\alpha}\log^{-\theta}(\frac{1}{x})$,
$\theta>\frac{1+3\alpha}{2}$, which improves on the the previously known bound, that $H^{\beta}(E) = 0$ for $\beta> \frac{1}{2}+\frac{3}{2}\alpha$. Further, by refining the argument in a subtle way, we are able to obtain a sharp dimension estimate for a whole class of {\em zero}-dimensional Furstenberg type sets. Namely, for
$\h_\gamma(x)=\log^{-\gamma}(\frac{1}{x})$, $\gamma>0$, we
construct a set $E_\gamma\in F_{\h_\gamma}$ of Hausdorff
dimension not greater than $\frac{1}{2}$.  Since in a previous work we showed that    $\frac{1}{2}$ is a lower bound  for the
Hausdorff dimension of any $E\in F_{\h_\gamma}$, with
the present construction, the value $\frac{1}{2}$ is sharp for
the whole class of Furstenberg sets associated to the zero
dimensional functions $\h_\gamma$.
\end{abstract}

\maketitle

\section{Introduction}\label{sec:intro}
We study dimension properties of sets of Furstenberg type. In
particular we are interested to be able to construct very small
Furstenberg sets in a given class. The notion of size of a set is
its Hausdorff dimension, denoted by $\dim_H$. We begin with
the definition of classical Furstenberg sets.

\begin{definition}
For $\alpha$ in $(0,1]$, a subset $E$ of $\RR$ is called
\textit{Furstenberg set} of type $\alpha$ or $F_\alpha$-set if
for each direction $e$ in the unit circle there is a line segment
$\ell_e$ in the direction of $e$ such that the Hausdorff
dimension of the set $E\cap\ell_e$ is greater or equal than
$\alpha$. We will also say that such set $E$ belongs to the class
$F_\alpha$.
\end{definition}

It is known (\cite{wol99b}, \cite{kt01}, \cite{mr10}) that $\dim_H(E)\ge\max\{2\alpha,\alpha+\frac{1}{2}\}$
for any $F_\alpha$-set $E\subseteq \RR$ and there are examples of
$F_\alpha$-sets $E$ with $\dim_H(E)\le
\frac{1}{2}+\frac{3}{2}\alpha$. Hence, if we denote by

\begin{equation*}
  \gamma(\alpha)=\inf \{\dim_H(E): E\in F_\alpha\},
\end{equation*}
then
\begin{equation}
\label{eq:dim}\max\left\{\alpha+\frac{1}{2} ; 2\alpha\right\}\le
\gamma(\alpha)\le\frac{1}{2}+\frac{3}{2}\alpha,\qquad
\alpha\in(0,1].
\end{equation}
In \cite{mr10} the left hand side of this inequality has been extended to the case of more general
dimension functions, i.e., functions that are not necessarily power functions
(\cite{Hau18}).

\subsection{Dimension Functions and Hausdorff measures}

\begin{definition}
A function $h$ will be called \textit{dimension function} if it
belongs to the following class:
\begin{equation*}
 \mathbb{H}:=\{h:[0,\infty)\to[0:\infty),
 \text{non-decreasing, right continuous, }
h(0)=0 \}.
\end{equation*}
\end{definition}

If one looks at the power functions, there is a natural
total order given by the exponents. If we denote by
$h_\alpha(x)=x^\alpha$, then $h_\alpha$ is {\em dimensionally
smaller} than $ h_\beta$ if and only if $\alpha<\beta$. In
$\mathbb{H}$, however, by extending this natural notion of order we
only obtain a \textit{partial} order.
\begin{definition}\label{def:order}
Let $g,h$ be two dimension functions. We will say that $g$ is
dimensionally smaller than $h$ and write $g\prec h $ if and only
if
    \begin{equation*}
    \lim_{x\to 0^+}\dfrac{h(x)}{g(x)}=0.
    \end{equation*}
\end{definition}

We will be particularly interested in the special subclass of
dimension functions that allows us to classify zero dimensional
sets.

\begin{definition}
A function $h\in\H$ will be called ``zero dimensional dimension
function'' if $h\prec x^\alpha$ for any $\alpha>0$. We denote by
$\mathbb{H}_0$ the subclass of those functions.
\end{definition}

As usual, the $h$-dimensional (outer) Hausdorff  measure $\HH{h}$
will be defined as follows. For a set $E\subseteq\R^n$ and
$\delta>0$, write
\begin{equation*}
    \hh{h}(E)=\inf\left\{\sum_i h(\diam(E_i)):E\subset\bigcup_i^\infty E_i,  \diam(E_i)<\delta \right\}.
\end{equation*}
The $h$-dimensional Hausdorff measure $\HH{h}$ of $E$ is defined
by
\begin{equation}
    \HH{h}(E)=\sup_{\delta>0 }\hh{h}(E). \label{eq:h-meas}
\end{equation}
We will consider functions $g,h$ such that there exist constants
$c,C$ with $0<c\le \frac{g(x)}{h(x)}\le C<\infty$ for all $x>0$
to be equivalent, even though they would not lead to the exact same measures. In that case we write $g\equiv h$.

 The Hausdorff dimension of a set $E\sub\R^n$ is the
unique real number $s$ characterized by the following properties:
\begin{itemize}
 \item $\HH{r}(E)=+\infty$ for all $r<s$.
 \item $\HH{t}(E)=0$ for all $s<t$.
\end{itemize}

Therefore, to prove that a given set   $E$ has dimension $s$, it is
enough to  check the preceding two properties, independently if
$\HH{s}(E)$ is zero, finite and positive, or infinite.

However,
in general it is not true that, given a set $E$, there is a
function $h\in\H$, such that if $g\succ h$ then
$\HH{g}(E)=0$, and if $g\prec h$, then $\HH{g}(E)=+\infty$.

The difficulties arise from two results due to Besicovitch  (see
\cite{rog98} and references therein). The first says that if a set
$E$ has null $\HH{h}$-measure for some $h\in\H$, then there
exists a function  $g$ which is dimensionally smaller than $h$ and for which still $\HH{g}(E)=0$.
Symmetrically, the second says that if a \emph{compact} set $E$
has non-$\sigma$-finite $\HH{h}$ measure, then there exists a
function $g\succ h$ such that $E$ has also non-$\sigma$-finite
$\HH{g}$ measure. These two results imply that if a set $E$
satisfies that there exists a function  $h$ such that
$\HH{g}(E)>0$ for any $g\prec h$ and $\HH{g}(E)=0$ for any
$g\succ h$, then it must be the case that
$0<\HH{h}(E)<+\infty$. This is the particular case in which $E$ is a so called $h$-set (generalizing the notion of
$s$-set for those $s$-dimensional sets that satisfy $0<H^s(E) < +\infty$.

 But if we look for example at the set $\LL$ of Liouville numbers, this set is not an $h$-set for any $h\in\H$. These sets are called
 {\em dimensionless} (see \cite{ek06}). In addition, it is shown in \cite{or06} that there are two proper nonempty subsets $\LL_0, \LL_\infty\sub\H$  of dimension functions such that $\HH{h}(\LL)=0$ for all $h\in\LL_0$ and $\HH{h}(\LL)=\infty$ for all $h\in\LL_\infty$. This motivates the following notion of dimension partition (see \cite{chm10}).

\subsection{Dimension Partition and Dimension Gaps}

\begin{definition}
By the \emph{Dimension Partition } \index{Dimension
functions!partition} of a set $E$ we mean a partition of $\H$
into (three) sets: $\mathcal{P}(E)=E_0\cup E_1\cup E_\infty$ with
\begin{itemize}
    \item $E_0=\{h\in\H:\HH{h}(E)=0\}$.
    \item $E_1=\{h\in\H:0<\HH{h}(E)<\infty\}$.
    \item $E_\infty=\{h\in\H:\HH{h}(E)=\infty\}$.
\end{itemize}
\end{definition}
Note that by the previous example, it is clear that there are sets
$E$ for which $E_1$ is empty, reflecting the dimensionless
nature of $E$. On the other hand, $E_1$ is never empty for an
$h$-set, but it is  not easy to determine this partition in the
general case. We also remark that it is possible to find
non-comparable dimension functions $g\not \equiv h$ and a set $E$ with the
property of being a $g$-set and an $h$-set simultaneously.

It follows that even for $h$-sets the dimension partition, and in
particular $E_1$, is not completely determined. Note that the
results of Besicovitch cited above imply that, for compact sets,
$E_0$ and $E_\infty$ can be thought of as open components of the
partition, and $E_1$ as the ``border'' of these open components.
An interesting problem is then to determine some criteria to
classify the functions in $\H$ into those classes.

To detect where this ``border'' is, we will introduce the notion
of \emph{chains} in $\H$. This notion allows to refine the notion
of Hausdorff dimension by using an ordered family of dimension
functions. More precisely, we have the following definition.

\begin{definition}\index{Dimension functions!chain}
A family $\mathcal{C}\subset\H$ of dimension functions will be
called a \emph{chain} if it is of the form
\begin{equation*}
\mathcal{C}=\left\{h_t\in\H: t\in\R, h_s\prec h_t \iff s<t\right\}.
\end{equation*}
That is, a totally ordered one-parameter family of dimension
functions.
\end{definition}

Suppose that $h\in\H$ belongs to some chain $\mathcal{C}$ and
satisfies that, for any $g\in\mathcal{C}$, $\HH{g}(E)>0$ if
$g\prec h$ and $\HH{g}(E)=0$ if $g\succ h$. Then, even if
$h\notin E_1$, in this chain, $h$ does measure the size of $E$.
It can be thought of as being ``near the frontier'' of both $E_0$
and $E_\infty$. For example, if a set $E$ has Hausdorff dimension
$\alpha$ but $\HH{\alpha}(E)=0$ or $\HH{\alpha}(E)=\infty$, take
$h_\alpha(x)=x^\alpha$ and $\mathcal{C}_H=\{x^t:t\ge 0\}$. In this
chain, $h_\alpha$ is the function that best measures the size of
$E$.

We look for finer estimates, considering chains of dimension
functions that separate sets of ``the same Hausdorff
dimension''. The goal will be to find very fine chains to give
precise bounds on how far from an expected dimension function the
Hausdorff measure drops to zero or remains positive. In this
setting, the sharpness of the results are associated to the
``fineness'' of the chains. Consider, for example, a set $A$ such
that $\dim_H(A)=\alpha$, in which case it is clear that
$\HH{\alpha+\e}(A)=0$ for all $\e>0$. Compare the chain
$\mathcal{C}_H$ (which only detects the Hausdorff dimension of
$A$) to
$\mathcal{C_{\text{log}}}=\{x^{\alpha}\log(\frac{1}{x})^{-\beta},
\beta\in\R \}$, in which we can detect if there exists
$\beta\in\R$ for which
$\HH{x^{\alpha}\log^{-\beta-\e}(\frac{1}{x})}(A)=0$ and
$\HH{x^{\alpha}\log^{-\beta+\e}(\frac{1}{x})}(A)=\infty$ for all
$\e>0$.

To measure this distance between to dimension functions, we
introduce the following notion:
\begin{definition}
 Let $g,h\in\H$ with $g\prec h$. Define the ``gap'' between $g$ and $h$ as
\begin{equation}
\Delta(g,h)(x)=\frac{h(x)}{g(x)}.
\end{equation}
\end{definition}
From this definition and the definition of partial order, we
always have that $\lim_{x\to 0}\Delta(g,h)(x)=0$, and therefore
the speed of convergence to zero of $\Delta(g,h)$ can be seen as a notion of
distance between $g$ and $h$.

\subsection{Generalized Furstenberg sets}

The analogous definition of Furstenberg sets in the setting of
dimension functions is the following.
\begin{definition}\label{def:furs}
Let $\h$ be a dimension function. A set $E\subseteq\RR$ is a
Furstenberg set of type $\h$, or an $F_\h$-set, if for each
direction $e\in\s$ there is a line segment $\ell_e$ in the
direction of $e$ such that  $\HH{\h}(\ell_e \cap E)>0$.
\end{definition}

In \cite{mr10} we proved that the appropriate dimension function
for an $F_\h$ set $E$ must be dimensionally not much smaller than $\h^2$
and $\h\sqrt{\cdot}$ (this is the generalized version of the left
hand side of \prettyref{eq:dim}), the latter with some additional
conditions on $\h$. There we also presented precise bounds on the
dimensional gaps.
These results are valid for all dimension functions $\h$ - minding the small restrictions on the decay of $\h$ for the second bound - regardless of the fact that $\h$ is (or is not) zero-dimensional. We also exhibited an example of a zero dimensional Furstenburg-type set with $2$ points in each direction, proving that for this case, $\inf{\dim E} = 0$ contradicting the na\"\i ve limiting argument in equation \prettyref{eq:dim}, that $\gamma(0) = \frac{1}{2}$.
(It seems feasible to find an argument to also construct a zero dimensional $F^K$-set for $3\leq K\leq \aleph_0$ points as remarked by Keleti and M\'ath\'e \cite{shm11})
However, if we consider zero-dimensional Furstenberg-type sets belonging to $F_{\h_\gamma}$, where
$\h_\gamma\in\H_0$ is defined by
$\h_\gamma(x)=\frac{1}{\log^\gamma(\frac{1}{x})}$, we showed that
any $E\in F_{\h_\gamma}$ must have $\dim_H(E)\ge\frac{1}{2}$.

In the present work we look at a refinement of the upper bound for the dimension of Furstenberg sets. Since we are looking for upper
bounds on a class of Furstenberg sets, the aim will be to explicitly construct  a very small set belonging to the given class.

We first consider the classical case of power functions,  $x^\alpha$, for $\alpha >0$. Recall that for this case,
the known upper bound implies that, for any positive
$\alpha$, there is a set $E\in F_\alpha$ such that
$\HH{\frac{1+3\alpha}{2}+\e}(E)=0$ for any $\e>0$. We improve
this estimate in terms of logarithmic gaps by showing that there
is no need to take a ``power like'' step from the critical
dimension. For any $\alpha>0$ consider
$h_\theta(x)=x^\frac{1+3\alpha}{2}\log^{-\theta}(\frac{1}{x})$.
Then, there is an $F_\alpha$-set $E$ such that
$\HH{\h_\theta}(E)=0$ for any $\theta>\frac{1+3\alpha}{2}$  (\prettyref{thm:logFalpha}). Note
that $x^\frac{1+3\alpha}{2}\prec h_\theta\prec
x^{\frac{1+3\alpha}{2}+\e}$ for all $\e>0$.

We will also focus at the endpoint $\alpha=0$, and give a
complete answer about the size of a class of Furstenberg sets by
proving that, for any given $\gamma>0$, there exists a set
$E_\gamma\sub\R^2$ such that (\prettyref{thm:sqrth3/2})
\begin{equation}\label{eq:smallmain}
 E_\gamma\in F_{\h_\gamma} \text{ for }\h_\gamma(x)=\frac{1}{\log^\gamma(\frac{1}{x})} \text{ and } \dim_H(E_\gamma)\le\frac{1}{2}.
\end{equation}
Note that, when $\gamma\to+\infty$, the dimension functions $\h_\gamma;\gamma>0$ increase, ``approaching'' from below the
positive-dimensional case. However, when $\gamma \to 0$, the functions decrease, ``approaching'' the
 sets having finitely many (or countably many) points in each direction. In this latter case, a zero-dimensional Furstenberg set can be constructed \cite{shm11}. We will provide an example of an even smaller Furstenberg set $E_{\h}$ (which naturally depends on the choice of $\h \in \H_0$) that has $2$ points in each direction, but such that $\HH{\h}(E_{\h}) = 0$  (see \prettyref{ex:F2Haus_h0}).

As usual, we will use the notation $A\lesssim B$ to indicate that
there is a constant $C>0$ such that $A\le C B$, where the
constant is independent of $A$ and $B$. By $A\sim B$ we mean that
both $A\lesssim B$ and $B\lesssim A$ hold.

The paper is organized as follows.  In \prettyref{sec:furs}, for a given $\alpha > 0$, we develop the main
construction of small Furstenberg ${\alpha}$ sets, and obtain dimension
estimates for the class $F_\alpha$, $\alpha>0$. In \prettyref{sec:gfurs} we first show that we can refine the argument of the previous section to include the zero-dimensional functions $\h_{\gamma}$ defined in \prettyref{eq:smallmain}. We then continue to improve on the construction given in \cite{mr10} to construct extremely small Furstenberg sets that contain 2 points in each direction and  we look at the same problem from the point of view of Packing dimension and compare the very different behaviour.  \prettyref{sec:fiber} is devoted to the proof of an important lemma regarding the size of zero dimensional fibers.
\section{Upper Bounds for Furstenberg-type Sets}\label{sec:furs}

In this section, given $\alpha >0$ we will focus our attention to the upper dimension estimate for $F_{\alpha}$
sets. As stated in \prettyref{eq:dim} it was shown that if $\gamma(\alpha)$ is the infimum of all the possible values
for the Hausdorff dimension of $F_\alpha$-sets, then
\begin{equation}\label{eq:dim_upper}
\max\{  2\alpha ;\frac{1}{2}+\alpha\}\le
\gamma(\alpha)\le\frac{1}{2}+\frac{3}{2}\alpha,\qquad
\alpha\in(0,1].
\end{equation}
We will concentrate on the right hand side of this inequality, which has been proved by showing that there exists a set $E$
 in $F_{\alpha}$, such that  $\HH{s}(E)=0$
for any $s>\frac{1+3\alpha}{2}$.

However, by the result of Besicovitch cited in the Introduction, we know that it is not going to be true that
$\HH{h}(E)=0$ for any $h\succ x^\frac{1+3\alpha}{2}$.

By refining the arguments of Wolff in \prettyref{thm:logFalpha}, if $\theta > \frac{1+3\alpha}{2}$ we are able to exhibit a set $E$ in $F_{\alpha}$, such that for $h_{\theta} := x^{\frac{1+3\alpha}{2}}\log^{-\theta}\left(\frac{1}{x}\right)$ we have that
$\HH{h_\theta}(E) = 0$.  Note that $h_\theta \prec x^s$ for any $s > \frac{1+3\alpha}{2}$.

We begin with a preliminary lemma about a very well distributed (mod 1) sequence.

\begin{lemma}\label{lem:discre}
 For $n\in \N$ and any real number $x\in [0,1]$, there is a pair $0\le j,k\le n-1$ such that
\begin{equation*}
\left|x-\left(\sqrt{2}\frac{k}{n}-\frac{j}{n}\right)\right|\le\frac{\log(n)}{n^2}.
\end{equation*}
\end{lemma}
This lemma is a consequence of Theorem 3.4 of \cite{kn74}, p125, in which an estimate is given about the discrepancy of the
fractional part of the sequence $\{n\alpha\}_{n\in\N}$ where $\alpha$ is a irrational of a certain type.

We also need to introduce the notion of $G$-sets, a common ingredient in the construction of Kakeya and Furstenberg sets.

\begin{definition}
 A $G$-set is a compact set $E\sub\R^2$ which is contained in the strip $\{(x,y)\in\R^2:0\le x \le 1\}$ such that for any $m\in[0,1]$ there is a line segment contained in $E$ connecting $x=0$ with $x=1$ of slope $m$, i.e.
\[
\forall m\in[0,1]\ \exists \ b\in\R: mx+b\in E,\ \forall \ x\in
[0,1].
\]
\end{definition}
Finally we need some notation for a thickened line.
\begin{definition}
 Given a line segment $\ell(x)=mx+b$, we define the $\delta$-tube associated to $\ell$ as
\[
S_\ell^\delta:=\{(x,y)\in\R^2:0\le x\le 1; |y-(mx+b)|\le\delta\}.
\]
\end{definition}\index{S@$S_\ell^\delta$, $\delta$-tube around the line $\ell$.}

Now we are ready to prove the main result of this section.

\begin{theorem}\label{thm:logFalpha}
For $\alpha\in(0,1]$ and $\theta>0$, define
$h_\theta(x)=x^{\frac{1+3\alpha}{2}}\log^{-\theta}(\frac{1}{x})$.
Then, if $\theta>\frac{1+3\alpha}{2}$, there exists a set
$E\in F_\alpha$ with $\HH{h_\theta}(E)=0$.
\end{theorem}

\begin{proof}

Fix $n\in\N$ and let $n_j$  be a sequence such that $n_{j+1}>n_j^j$. We consider $T$ to be the set defined as follows:
\begin{equation*}
T=\left\{x\in\left[\frac{1}{4},\frac{3}{4}\right]: \forall j \
\exists\ p,q\ ; q\le n_j^\alpha;
|x-\frac{p}{q}|<\frac{1}{n_j^2}\right\}.
\end{equation*}

It can be seen that $\dim_H(T)=\alpha$ (see \prettyref{sec:fiber}, \prettyref{thm:jarnikA}).

If $\varphi(t)=\frac{1-t}{t\sqrt{2}}$ and $D=\varphi^{-1}\left([\frac{1}{4},\frac{3}{4}]\right)$, we have that $\varphi:D\to [\frac{1}{4},\frac{3}{4}]$ is bi-Lipschitz. Therefore the set
\begin{equation*}
T'=\left\{t\in\R:\frac{1-t}{t\sqrt{2}}\in
T\right\}=\varphi^{-1}(T)
\end{equation*}
also has Hausdorff dimension $\alpha$.

The main idea of our proof, is to construct a set for which we have,
essentially, a copy of $T'$ in each direction and simultaneously
keep some optimal covering property.

Define, for each $n\in \N$,
\begin{equation*}
\Gamma_n:=\left\{\frac{p}{q}\in\left[\frac{1}{4},\frac{3}{4}\right],
q\le n^\alpha\right\}
\end{equation*}
and
\begin{equation*}
Q_n=\left\{ t:\frac{1-t}{\sqrt{2}t}=\frac{p}{q}\in
\Gamma_n\right\}=\varphi^{-1}(\Gamma_n).
\end{equation*}

To count the elements of $\Gamma_n$ (and $Q_n$), we take into
account that
\begin{equation*}
\sum_{j=1}^{\lfloor n^{\alpha}\rfloor}j\le \frac{1}{2} \lfloor
n^{\alpha}\rfloor(\lfloor n^{\alpha}\rfloor+1) \lesssim \lfloor
n^{\alpha}\rfloor^2\le n^{2\alpha}.
\end{equation*}
Therefore, $\#(Q_n)\lesssim n^{2\alpha}$.

For $0\le j,k\le n-1$, define the line segments
\[
\ell_{jk}(x):=(1-x)\frac{j}{n}+x\sqrt{2}\frac{k}{n} \text{ for }
x\in[0,1],
\]
and their $\delta_n$-tubes $S_{\ell_{jk}}^{\delta_n}$ with
$\delta_n=\frac{\log(n)}{n^2}$. We will use during the proof the
notation $S_{jk}^n$ instead of $S_{\ell_{jk}}^{\delta_n}$. Also
define
\begin{equation}\label{eq:Gn}
G_n:=\bigcup_{jk}S_{jk}^n.
\end{equation}
Note that, by \prettyref{lem:discre}, all the $G_n$ are $G$-sets.

For each $t\in Q_n$, we look at the points $\ell_{jk}(t)$, and
define the set $S(t):=\{\ell_{jk}(t)\}_{j,k=1}^n$. Clearly, $\#(S(t))\le
n^2$. But if we note that, if $t\in Q_n$, then
\begin{equation*}
0\le\frac{\ell_{jk}(t)}{t\sqrt{2}}=\frac{1-t}{t\sqrt{2}}\frac{j}{n}+\frac{k}{n}=\frac{p}{q}\frac{j}{n}+\frac{k}{n}=\frac{pj+kq}{nq}<2,
\end{equation*}
we can bound $\#(S(t))$ by the number of non-negative rationals
smaller than 2 of denominator $qn$. Since $q\le n^\alpha$, we
have $\#(S(t))\le n^{1+\alpha}$. Considering \emph{all} the
elements of $Q_n$, we obtain $\#\left(\bigcup_{t\in
Q_n}S(t)\right)\lesssim n^{1+3\alpha}$. Let us define
\begin{equation*}
\Lambda_n:=\left\{(x,y)\in G_n: |x-t|\le \frac{C}{n^2}\text{ for
some } t\in Q_n\right\}.
\end{equation*}

\begin{claim}
 For each $n$, take $\delta_n=\frac{\log(n)}{n^2}$. Then $\Lambda_n$ can be covered by $L_n$ balls of radio $\delta_n$ with $L_n \lesssim n^{1+3\alpha}$.
\end{claim}
To see this, it suffices to set a parallelogram on each point of
$S(t)$ for each $t$ in $Q_n$. The lengths of the sides of the
parallelogram are of order $n^{-2}$ and $\frac{\log(n)}{n^2}$, so
their diameter is bounded by  a constant times
$\frac{\log(n)}{n^2}$, which proves the claim.

We   can now begin with the recursive construction that leads to the
desired set. Let $F_0$ be a $G$-set written as
\begin{equation*}
F_0=\bigcup_{i=1}^{M_0}S_{\ell_i^0}^{\delta^0},
\end{equation*}
(the union of $M_0$ $\delta^0$-thickened line segments
$\ell^0_i=m^0_i+b^0_i$ with appropriate orientation). Each $F_j$
to be constructed will be a $G$-set of the form
\begin{equation*}
F_j:=\bigcup_{i=1}^{M_j}S_{\ell_i^j}^{\delta^j}, \qquad \text{
with } \ \ell^j_i=m^j_i+b^j_i.
\end{equation*}
 Having constructed $F_j$, consider the $M_j$ affine mappings
\begin{equation*}
A^j_i:[0,1]\times[-1,1]\rightarrow
S_{\ell^j_{i}}^{\delta^j}\qquad 1\le i\le M_j,
\end{equation*}
defined by
\begin{equation*}
  A^j_i\left(\begin{array}{c}
    x\\
    y
\end{array}\right)=\left(\begin{array}{cc}
    1 & 0\\
    m_i^j & \delta^j
\end{array}\right)\left(\begin{array}{c}
    x\\
    y
\end{array}\right)+\left(\begin{array}{c}
    0\\
    b^j_{i}
\end{array}\right).
\end{equation*}
We choose $n_{j+1}$ large enough to satisfy
\begin{equation}\label{eq:njloglogMj}
\log\log(n_{j+1})>M_j
\end{equation}
and apply $A_i^j$ to the sets $G_{n_{j+1}}$ defined in
\prettyref{eq:Gn} to obtain
\begin{equation*}
F_{j+1}=\bigcup_{i=1}^{M_j}A_i^j(G_{n_{j+1}}).
\end{equation*}
Since $G_{n_{j+1}}$ is a union of thickened line segments, we
have that
\begin{equation*}
F_{j+1}=\bigcup_{i=1}^{M_{j+1}}S_{\ell_i^{j+1}}^{\delta^{j+1}},
\end{equation*}
for an appropriate choice of $M_{j+1}$, $\delta_{j+1}$ and
$M_{j+1}$ line segments $\ell_i^{j+1}$. From the definition of
the mappings $A_i^j$ and since the set $G_{n_{j+1}}$ is a
$G$-set, we conclude that $F_{j+1}$ is also a $G$-set. Define
\begin{equation*}
E_j:=\{(x,y)\in F_j: x\in T'\}.
\end{equation*}
To cover $E_j$, we note that if $(x,y)\in E_j$, then $x\in T'$,
and therefore there exists a rational $\frac{p}{q}\in
\Gamma_{n_j}$ with
\begin{equation*}
\frac{1}{n_j^2}>\left|\frac{1-x}{x\sqrt{2}}-\frac{p}{q}\right|=|\varphi(x)-\varphi(r)|\ge\frac{|x-r|}{\sqrt{2}},\qquad
\text{ for some }\ r\in Q_{n_j}.
\end{equation*}
Therefore $(x,y)\in
\bigcup_{i=1}^{M_{j-1}}A^{j-1}_i(\Lambda_{n_j})$, so we conclude
that $E_j$ can be covered by $M_{j-1}n_{j}^{1+3\alpha}$ balls of
diameter at most $\frac{\log(n_{j})}{n_{j}^2}$. Since we chose
$n_j$ such that $\log\log(n_j)>M_{j-1}$, we obtain that $E_{j}$
admits a covering by $\log\log(n_{j})n_{j}^{1+3\alpha}$ balls of
the same diameter. Therefore, if we set $F=\bigcap_j F_j$ and
$E:=\{(x,y)\in F: x\in T'\} $ we obtain that
\begin{eqnarray*}
\HH{h_\theta}_{\delta_j}(E) & \lesssim  & n^{1+3\alpha}_{j}\log\log(n_j)h_\theta\left(\frac{\log(n_j)}{n^2_j}\right)\\
    & \lesssim & n^{1+3\alpha}_{j}\log\log(n_j)\left(\frac{\log(n_j)}{n^2_j}\right)^{\frac{1+3\alpha}{2}}\log^{-\theta}\left(\frac{n^2_j}{\log(n_j)}\right)\\
    & \lesssim & \log\log(n_j)\log(n_j)^{\frac{1+3\alpha}{2}-\theta}\lesssim  \log^{\frac{1+3\alpha}{2}+\e-\theta}(n_j)
\end{eqnarray*}
for $x\ge K=K(\e)$. Therefore, for any
$\theta>\frac{1+3\alpha}{2}$, the last expression goes to zero.
In addition, $F$ is a $G$-set, so it must contain a line segment
in each direction $m\in[0,1]$. If $\ell$ is such a line segment,
then
\begin{equation*}
\dim_H(\ell\cap E)=\dim_H(T')\ge\alpha.
\end{equation*}
The final set of the proposition is obtained by taking eight
copies of $E$, rotated to achieve \emph{all} the directions in
$\s$.
\end{proof}

\section{Upper Bounds for small Furstenberg-type Sets}\label{sec:gfurs}

In this section we will focus on the class $F_\alpha$ at the
endpoint $\alpha=0$. Note that all preceding results involved only the case for which $\alpha > 0$. Introducing the generalized Hausdorff measures, we are able to handle an important class of Furstenberg type sets in $F_0$.

We will  look first at the $0$-dimensional case, but in which we request the set to contain uncountably many points.
For this, it will be necessary to put in each direction some set with many points but with certain structure.

THe proof relies on \prettyref{thm:logFalpha}, but we must replace the set $T$ by a generalized version of it. More
precisely, we will need the following lemma.

\begin{lemma}\label{lem:fiber}
Let $r>1$ and consider the sequence $\n=\{n_j\}$ defined by $n_j=e^{\frac{1}{2}n_{j-1}^{\frac{4}{r}j}}$, the function
$\f(x)=\log(x^2)^\frac{r}{2}$ and the set
\begin{equation*}
T=\left\{x\in\left[\frac{1}{4},\frac{3}{4}\right]\setminus \Q: \forall j \ \exists\ p,q\ ; q\le
\f(n_j); |x-\frac{p}{q}|<\frac{1}{n_j^2}\right\}.
\end{equation*}
Then we have that $\HH{\h}(T)>0$ for $\h(x)=\frac{1}{\log(\frac{1}{x})}$.
\end{lemma}

We postpone the proof of this lemma to the next section. With this lemma, we are able to prove the main result of this section. We have the next theorem.

\begin{theorem}\label{thm:sqrth3/2}
Let $\h=\frac{1}{\log(\frac{1}{x})}$. There exists a set $E\in
F_\h$ such that $\dim_H(E)\le \frac{1}{2}$.
\end{theorem}

\begin{proof}
We will use essentially a copy of $T$ in each direction in the construction of the desired set to fulfill the conditions required to be an $F_\h$-set. Let $T$ be the set defined in \prettyref{lem:fiber}. Define $T'$ as
\begin{equation*}
T'=\left\{t\in\R:\frac{1-t}{t\sqrt{2}}\in
T\right\}=\varphi^{-1}(T),
\end{equation*}
where $\varphi$ is the same bi-Lipschitz function from the proof of \prettyref{thm:logFalpha}. Then $T'$ has positive $\HH{\h}$-measure. Let us define the
corresponding sets of \prettyref{thm:logFalpha} for this generalized case.
\begin{equation*}
\Gamma_n:=\left\{\frac{p}{q}\in\left[\frac{1}{4},\frac{3}{4}\right],
q\le \f(n)\right\},
\end{equation*}
\begin{equation*}
Q_n=\left\{ t:\frac{1-t}{\sqrt{2}t}=\frac{p}{q}\in
\Gamma_n\right\}=\varphi^{-1}(\Gamma_n).
\end{equation*}
Now the estimate is $\#(Q_n)\lesssim
\f^{2}(n)=\log^r(n^2)\sim\log^r(n)$, since
\begin{equation*}
\sum_{j=1}^{\lfloor \f(n)\rfloor}j\le \frac{1}{2} \lfloor
\f(n)\rfloor(\lfloor \f(n)\rfloor+1) \lesssim \lfloor
\f(n)\rfloor^2.
\end{equation*}
For each $t\in Q_n$, define $S(t):=\{\ell_{jk}(t)\}_{j,k=1}^n$.
If $t\in Q_n$, following the previous ideas, we obtain that
\begin{equation*}
\#(S(t))\lesssim n\log^\frac{r}{2}(n),
\end{equation*}
and therefore
\begin{equation*}
\#\left(\bigcup_{t\in Q_n}S(t)\right)\lesssim
n\log(n)^{\frac{3r}{2}}.
\end{equation*}
Now we estimate the size of a covering of
\begin{equation*}
\Lambda_n:=\left\{(x,y)\in G_n: |x-t|\le \frac{C}{n^2}\text{ for
some } t\in Q_n\right\}.
\end{equation*}
For each $n$, take $\delta_n=\frac{\log(n)}{n^2}$. As before, the
set $\Lambda_n$ can be covered with  $L_n$ balls of radio
$\delta_n$ with $L_n\lesssim n\log(n)^\frac{3r}{2}$.

Once again, define $F_j$, $F$, $E_j$ and $E$ as before. Now the
sets $F_j$ can be covered by less than
$M_{j-1}n_j\log(n_j)^\frac{3r}{2}$ balls of diameter at most
$\frac{\log(n_{j})}{n_{j}^2}$. Now we can verify that, since each
$G_n$ consist of $n^2$ tubes, we have that $M_j=M_0n_1^2\cdots
n_j^2$. We can also verify that the sequence $\{n_j\}$ satisfies
the relation $\log{n_{j+1}}\ge M_{j}=M_0n_1^2\cdots n_j^2$, and
therefore we have the bound

\begin{equation*}
\dim_H(E)\le\underline{\dim}_B(E)\le\varliminf_j\frac{\log\left(\log(n_{j})
n_j\log(n_j)^\frac{3r}{2}
\right)}{\log\left(n_{j}^2\log^{-1}(n_j)\right)}=\frac{1}{2},
\end{equation*}
 where $\underline{\dim}_B$ stands for the lower box dimension.
Finally, for any $m\in[0,1]$ we have a line segment $\ell$ with
slope $m$ contained in $F$. It follows that $\HH{\h}(\ell\cap
E)=\HH{\h}(T')>0$.
\end{proof}

We remark that the argument in this particular result is
essentially the same needed to obtain the family of Furstenberg
sets $E_\gamma\in F_{\h_\gamma}$ for
$\h_\gamma(x)=\frac{1}{\log^\gamma(\frac{1}{x})}$,
$\gamma\in\R_+$, such that $\dim_H(E_\gamma)\le \frac{1}{2}$
announced in the introduction.

 We will now look at the case of finite number of points in each direction.

\subsection{The case \texorpdfstring{$\alpha=0$}{a=0}, \texorpdfstring{$K$}{k} points}
Let us begin with the definition of the class $F^K$.
\begin{definition}\label{def:FK}
For $K\in\N$, $K\ge 2$, a set will be a $F^K$-set or a
Furstenberg set of type $K$ if for any direction $e\in\s$, there
are at least $K$ points contained in $E$ lined up in the
direction of $e$.
\end{definition}

This case contrasts with the previous case, since we show that in this case $\frac{1}{2}$ is not longer the smallest possible
dimension. Already in \cite{mr10} we proved that there is a $F^2$-set with zero Hausdorff dimension.
We will generalize this example to obtain even smaller $F^2$ sets. Namely, for any $h\in\H_0$, there exists $G$ in $F^2$ such that $\HH{h}(G)=0$.
It is clear that  the set $G$ will depend on the choice of $h$.

\begin{example}\label{ex:F2Haus_h0}

Given a function $h\in\H$, we will construct two small sets $E, F\sub[0,1]$ with $\HH{h}(E)=\HH{h}(F)=0$ and such that $[0,1]\sub E+F$. Consider now $G=E\times \{1\}\cup -F\times\{0\}$. Clearly, we have that $\HH{h}(G)=0$, and contains two points in every direction $\theta\in [0;\frac{\pi}{4}]$. For, if
$\theta\in [0;\frac{\pi}{4}]$, let $c=\tan(\theta)$, so $c\in[0,1]$. By the choice of $E$ and  $F$, we can find $x\in E$ and $y\in F$ with $c=x+y$. The points $(-y,0)$ and $(x,1)$ belong to $G$ and determine a segment in the direction $\theta$.

We define
\begin{equation*}
    E:=\left\{x\in [0,1]: r_j=0 \text{ if } m_k+1\le j\le m_{k+1}; k \text{ even} \right\}
\end{equation*}
and
\begin{equation*}
    F:=\left\{x\in [0,1]: r_j=0 \text{ if } m_k+1\le j\le m_{k+1}; k \text{ odd } \right\}.
\end{equation*}
Here $\{m_k; m_0=0\}_k$ is an increasing sequence such that
$m_k\to+\infty$.  Now we estimate the size of the set $E$. Given
$k\in\N$, $k$ even, define $\ell_k=m_k-m_{k-1}+\dots+m_2-m_1$. It
is clear that $E$ can be covered by $2^{l_k}$ intervals of length
$2^{-m_{k+1}}$. Therefore, if the sequence $m_k$ increases fast
enough, then
\begin{equation*}
 \dim_H(E)\le\lbd(E)\le\varliminf_k \frac{\log(2^{\ell_k})}{\log (2^{m_{k+1}})}\lesssim\varliminf_k \frac{2^{\ell_k}}{2^{m_{k+1}}}=0.
\end{equation*}
Since the same argument shows that $\dim_H(F)=0$, this estimate
proves that the set $G$ has Hausdorff
dimension equal to zero. Now, for the finer estimate on the
$\HH{h}$-measure of the set, we must impose a more restrictive
condition on the sequence $\{m_k\}$.

Recall that the covering property implies that, for a given
$h\in\H$, we have that
\begin{equation*}
 \HH{h}(E)\le 2^{l_k}h(2^{-m_{k+1}}).
\end{equation*}
Therefore we need to choose a sequence $\{m_j\}$, depending on
$h$, such that the above quantity goes to zero with $k$. Since
$\ell_k\le m_k$, we can define recursively the sequence $\{m_k\}$
to satisfy the relation
\begin{equation*}
 2^{m_k}h(2^{-m_{k+1}})= \frac{1}{k}.
\end{equation*}
This last condition is equivalent to $
m_{k+1}=\log\left(\frac{1}{h^{-1}(\frac{1}{k2^{m_k}})}\right)$.
As an concrete example, take $h(x)=\frac{1}{\log(\frac{1}{x})}$.
In this case we obtain that the sequence $\{m_k\}$ can be defined
as
\begin{equation*}
 m_{k+1}=k2^{m_k}.
\end{equation*}
\end{example}

\subsection{Remark about the Packing dimension for small sets}
It is worthy to note here that if we were to measure the size of
Furstenberg sets with the packing dimension, the situation is
absolutely different. More precisely, for $K\ge 2$, any $F^K$-set
$E\subset \RR$ must have $\dim_P(E)\ge \frac{1}{2}$. For, if $E$
is an $F^2$ set, then the map $\varphi$ defined by
 $\varphi(a,b)=\frac{a-b}{\|a-b\|}$
is Lipschitz when restricted to $G_\e:=E\times
E\setminus\{(x,y)\in E\times E: \|(x,y)-(a,a)\|<\e; a\in E\}$.
Roughly, we are considering the map that recovers the set
of directions but restricted ``off the diagonal''. It is clear
that we can assume without loss of generality that all the pairs
are the endpoints of unit line segments. Therefore, since $E$ is
an $F^K$-set, $\varphi(G_\e)=\s$ if $\e$ is small enough. We
obtain the inequality
\begin{equation*}
 1=\dim_H(\s)\le \dim_H(G_\e)\le\dim_H(E\times E).
\end{equation*}
The key point is the product formulae for Hausdorff and Packing
dimensions. We obtain that
\begin{equation}\label{eq:haus+packge1}
 1\le\dim_H(E\times E)\le \dim_H(E)+\dim_P(E)\le 2\dim_P(E)
\end{equation}
 and then $\dim_P(E)\ge\frac{1}{2}$. It also follows that if we achieve small Hausdorff dimension then the Packing dimension is forced to increase. In particular, the $F^2$-set constructed in \cite{mr10} has Hausdorff dimension 0 and therefore it has Packing dimension 1.

 By means of an example we illustrate the relation between Hausdorff and Packing dimension for Furstenberg sets. It can be understood as optimal in the
sense of obtaining the smallest possible dimensions, both
Hausdorff and Packing.
Recall that from \prettyref{eq:haus+packge1} we have that for any $K\ge2$, any Furstenberg set $E$ of the class $F^K$ must satisfy
\begin{equation*}
1\le \dim_H(E)+\dim_P(E) \le 2\dim_P(E).
\end{equation*}
In fact, there is an $F^2$ set $E$ such that $\dim_H(E)=\frac{1}{2}=\dim_P(E)$.
\begin{example}\label{ex:F2Haus=pack=1/2}
The construction is essentially the same as in \prettyref{ex:F2Haus_h0}, but we use two
different sets to obtain all directions. Let $A$ be the set of
all the numbers whose expansion in base $4$ uses only the digits
$0$ and $1$. On the other hand, let $B$ the set of those numbers
which only uses the digits $0$ and $2$. Both sets have Packing
and Hausdorff dimension equal to $\frac{1}{2}$ and
$[0,1]\subseteq A+B$. The construction follows then the same pattern
as in the previous example.
\end{example}

\section{Proof of \prettyref{lem:fiber}}\label{sec:fiber}

The purpose of this section is to prove \prettyref{lem:fiber}. Our proof relies on a variation of a Jarn\'ik type theorem on Diophantine approximation.  We begin with some preliminary results on Cantor type constructions that will be needed.

\subsection{Cantor sets}\label{sec:cantor}

In this section we introduce the construction of sets of Cantor
type in the spirit of \cite{fal03}. By studying two quantities,
the number of children of a typical interval and some separation
property, we obtain sufficient conditions on these quantities
that imply the positivity of the $h$-dimensional measure for a
test function $h\in\H$.

We will need a preliminary elemental lemma about concave
functions. The proof is straightforward.
\begin{lemma}\label{lem:lemaconcava}
Let $h\in \H$ be a concave dimension function. Then
\[
 \min\{a,b\}\le \frac{a}{h(a)}h(b)\quad\text{ for any } a,b\in\R_+.
\]
\end{lemma}

\begin{proof}
We consider two separate cases:
\begin{itemize}
    \item If $b\ge a$ then $\frac{a}{h(a)}h(b)\ge\frac{a}{h(a)}h(a)=a=\min\{a,b\}$.
    \item If $a> b$, then $\frac{a}{h(a)}h(b)\ge\frac{b}{h(b)}h(b)=b=\min\{a,b\}$ by concavity of $h$.
\end{itemize}
\end{proof}
The following lemma is a natural extension of the ``Mass
Distribution Principle'' to the dimension function setting.
\begin{lemma}[$h$-dimensional mass distribution principle]\label{lem:massh}
 Let $E\sub\R^n$ be a set, $h\in\H$ and $\mu$ a probability measure on $E$. Let $\e>0$ and $c>0$ be positive constants such that for any $U\sub\R^n$ with $\emph{\diam}(U)<\e$ we have
\begin{equation*}
 \mu(U)\le c h(\emph{\diam}(U)).
\end{equation*}
Then $\HH{h}(E)>0$.
\end{lemma}
\begin{proof}
For any $\delta$-covering we have
\begin{equation*}
 0<\mu(E)\le \sum_i\mu(U)\le c\sum_i h(\diam(U)).
\end{equation*}
Then $\hh{h}>\frac{\mu(E)}{c}$ and therefore $\HH{h}(E)>0$.
\end{proof}
Now we present the construction of a Cantor-type set (see Example 4.6 in \cite{fal03}).
\begin{lemma}\label{lem:cantorgeneral}
Let $\{E_k\}$ be a decreasing sequence of closed subsets of the
unit interval. Set $E_0=[0,1]$ and suppose that the following
conditions are satisfied:
\begin{enumerate}
    \item Each $E_k$ is a finite union of closed intervals $I^k_j$.
    \item \label{eq:mk} Each level $k-1$ interval contains at least $m_k$ intervals of level $k$. We will refer to this as the ``children'' of an interval.
    \item The gaps between the intervals of level $k$ are at least of size $\e_k$, with $0<\e_{k+1}<\e_k$.
\end{enumerate}
Let $E=\bigcap_k E_k$. Define, for a \textbf{concave} dimension
function $h\in\H$, the quantity
\begin{equation*}
D^h_k:=m_1\cdot m_2\cdots m_{k-1}h(\e_km_k).
\end{equation*}
If $\varliminf_k D^h_k>0$, then $\HH{h}(E)>0$.
\end{lemma}

\begin{proof}
The idea is to use the version of the mass distribution principle
from \prettyref{lem:massh}. Clearly we can assume that the
property \prettyref{eq:mk} of \prettyref{lem:cantorgeneral} holds
for \emph{exactly} $m_k$ intervals. So we can define a mass
distribution on $E$ assigning a mass of $\frac{1}{m_1\cdots m_k}$
to each of the  $m_1\cdots m_k$ intervals of level $k$. Now, for
any interval $U$ with $0<|U|<\e_1$, take $k$ such
$\e_k<|U|<\e_{k-1}$. We will estimate the number of intervals of
level $k$ that could have non-empty intersection with $U$. For
that, we note the following:
\begin{itemize}
    \item $U$ intersects at most one $I^{k-1}_{j}$, since $|U|<\e_{k-1}$. Therefore it can intersect at most $m_k$ children of $I^{k-1}_{j}$.
    \item Suppose now that $U$ intersects $L$ intervals of level $k$. Then it must contain $(L-1)$ gaps of size at least $\e_k$. Therefore, $L-1\le\frac{|U|}{\e_k}$. Consequently $|U|$ intersects at most $\frac{|U|}{\e_k}+1\le 2\frac{|U|}{\e_k}$ intervals of level $k$.
\end{itemize}
From these two observations, we conclude that
\begin{equation*}
\mu(U)\le\frac{1}{m_1\cdots m_k}\min\left\{m_k,
\frac{2|U|}{\e_k}\right\}=\frac{1}{m_1\cdots
m_k\e_k}\min\{\e_km_k,2|U|\}.
\end{equation*}
Now, by the concavity of $h$, we obtain
\begin{equation*}
 \min\{\e_km_k,2|U|\}\le \frac{\e_km_k}{h(\e_km_k)}h(2|U|).
\end{equation*}
In addition (also by concavity), $h$ is doubling, so
$h(2|U|)\lesssim h(|U|)$ and then
\begin{equation*}
\mu(U)\lesssim\frac{\e_km_kh(|U|)}{m_1\cdots
m_k\e_kh(\e_km_k)}=\frac{h(|U|)}{m_1\cdots
m_{k-1}h(\e_km_k)}=\frac{h(|U|)}{D^h_k}.
\end{equation*}
Finally, if $\varliminf_k D_k^h>0$, there exists $k_0$ such
$\frac{1}{D_k^h}\le C$ for $k\ge k_0$ and we can use the mass
distribution principle with $C$ and $\e=\e_{k_0}$.
\end{proof}

\begin{remark}
In the particular case of $h(x)=x^s$, $s\in(0,1)$ we recover the
result of \cite{fal03}, where the parameter $s$ can be expressed
in terms of the sequences $m_k$ and $\e_k$. For the set
constructed in \prettyref{lem:cantorgeneral}, we have
\begin{equation}\label{eq:classicgeneral}
 \dim_H(E)\ge \frac{\log(m_1\cdots m_{k-1})}{-\log(m_k\e_k)}.
\end{equation}
\end{remark}

\subsection{Diophantine approximation - Jarn\'ik's Theorem}\label{sec:dioph}

The central problem in the theory of Diophantine approximation is, at its simplest level, to approximate irrational numbers by rationals. A classical theorem due to Jarn\'ik in this area is the following (see \cite{fal86}), which provides a result on the size of the set of real numbers that are well approximable:
\begin{theorem}\label{thm:jarnikB}
For $\beta\ge2$, define the following set:
\[
B_\beta=\left\{x\in[0,1]\setminus \Q:
\|xq\|<\frac{1}{q^{\beta-1}}\text{ for infinitely many }
q\in\Z\right\}.
\]
Then $\dim_H(B_\beta)=\frac{2}{\beta}$.
\end{theorem}

In fact, there are several results (see \cite{khi24}, \cite{jar31}) regarding the more general problem of estimating the dimension partition for the set
\begin{equation}\label{eq:Bg}
B_\g:=\left\{x\in[0,1]\setminus \Q:
\|xq\|\le\frac{q}{\g(q)}\text{ for infinitely many }
q\in\N\right\},
\end{equation}
where $\g$ is any positive increasing function.

We can therefore see \prettyref{lem:fiber} as a result on the size of a set of well approximable numbers. We will derive the proof from the following proposition, where we prove a lower bound estimate for the set $B_\g$. Precisely, for $\h(x)=\frac{1}{\left(\g^{-1}(\frac{1}{x})\right)^2}$ we will find conditions on $\Delta(h,\h)=\frac{\h}{h}$  with $h\prec \h$ to ensure that $\HH{h}(B_\g)>0$.

\begin{proposition}\label{pro:Bglow}
Let $\g$ be a positive, increasing function satisfying
\begin{equation}\label{eq:fastg}
\g(x)\gg x^2 \quad  (x\gg 1)
\end{equation}
and
\begin{equation}\label{eq:logg-1}
\g^{-1}(ab)\lesssim \g^{-1}(a)+\g^{-1}(b) \text{ for all }
a,b\ge1.
\end{equation}
Define $B_\g$ as in \prettyref{eq:Bg} and let $h\in\H_d$
such that $h\prec\h(x)=\frac{1}{\left(\g^{-1}(\frac{1}{x})\right)^2}$. Consider a sequence $\{n_k\}$ that satisfies:
\begin{itemize}
\item[(A)] $n_{k}\ge 3\g(2n_{k-1})$.
\item[(B)] $\log(n_k)\le \g(n_{k-1})$.
\end{itemize}
If
$\Delta(h,\h)(x)=\frac{\h(x)}{h(x)}=\frac{1}{h(x)\g^{-1}(\frac{1}{x})^2}$
satisfies
\begin{equation}\label{eq:critlow}
\varliminf_k\frac{1}{6^k\g^{2}(n_{k-2})\Delta(h,\h)\left(\frac{1}{\log(n_{k})\g(n_{k-1})}\right)}>0,
\end{equation}
then $\HH{h}(B_\g)>0$.
\end{proposition}
\begin{proof}
Define
\begin{equation}
 G_q:=\{x\in [0,1]\setminus \Q: \|xq\|\le \frac{q}{\g(q)}\}.
\end{equation}
For each $q\in\N$, $G_q$ is the union of $q-1$ intervals of
length $2\g(q)^{-1}$ and two more intervals of length
$\g(q)^{-1}$ at the endpoints of $[0,1]$:
\begin{equation*}
G_q=\left(0,\frac{1}{\g(q)}\right) \cup
\left(\frac{1}{\g(q)},1\right) \cup \bigcup_{1\le r \le
q-1}\left(\frac{r}{q}-\frac{1}{\g(q)},\frac{r}{q}+\frac{1}{\g(q)}\right)
\end{equation*}

Now define for each $q$
\[
G'_q:=G_q\cap \left(\frac{1}{\g(q)},1-\frac{1}{\g(q)}\right).
\]

Now, for each $n\in\N$ consider two prime numbers $p_1,p_2$ such
that $n\le p_1<p_2<2n$. We will prove that $G'_{p_1}$ and
$G'_{p_2}$ are disjoint and well separated. Note that if
$\frac{r_1}{p_1}$ and $\frac{r_2}{p_2}$ are centers of two of the
intervals belonging to $G'_{p_1}$ and $G'_{p_2}$, we have
\[
 \left|\frac{r_1}{p_1}-\frac{r_2}{p_2}\right|=\frac{1}{p_1p_2}|r_1p_2-r_2p_1|\ge\frac{1}{4n^2},
\]
since $r_1p_2-r_2p_1\neq 0$. Therefore, taking into account this
separation between the centers and the length of the intervals,
we conclude that for $x\in G'_{p_1}$ and $y\in G'_{p_2}$,
\[
 |x-y|\ge \frac{1}{4n^2}-\frac{2}{\g(n)}\ge \frac{1}{8n^2}\qquad(\text{ since }\g(n)\gg n^2).
\]
Let $\mathcal{P}_m^n$ be the set of all the prime numbers between
$m$ and $n$ and define
\[
    H_n:=\bigcup_{p\in\mathcal{P}_n^{2n}}G'_p.
\]
Then $H_n$ is the union of intervals of length at least
$\frac{2}{\g(2n)}$ that are separated by a distance of at least
$\frac{1}{8n^2}$.

Now we observe the following: If $I$ is an interval with
$|I|>\frac{3}{n}$, then at least $\frac{p|I|}{3}$ of the
intervals of $G'_p$ are completely contained on $I$. To verify
this last statement, cut $I$ into three consecutive and congruent
subintervals. Then, in the middle interval there are at least
$\frac{p|I|}{3}$ points of the form $\frac{m}{p}$. All the
intervals of $G'_p$ centered at these points are completely
contained in $I$, since the length of each interval of $G'_p$ is
$\frac{2}{\g(p)}<\frac{|I|}{3}$.

In addition, by the Prime Number Theorem, we know that
$\#(\mathcal{P}_1^n)\sim\frac{n}{\log(n)}$, so we can find $n_0$
such that
\begin{eqnarray*}
\#(\mathcal{P}_n^{2n}) \ge \frac{n}{2\log(n)} \text{ for } n\ge
n_0.
\end{eqnarray*}

Hence, if $I$ is an interval with $|I|>\frac{3}{n}$, then there
are at least
\[
\frac{p|I|}{3}\frac{n}{2\log(n)}>\frac{n^2|I|}{6\log(n)}
\]
intervals of $H_n$ contained on $I$. Now we will construct a
Cantor-type subset $E$ of $B_\g$ and apply
\prettyref{lem:cantorgeneral}.

Consider the sequence $\{n_k\}$ of the hypothesis of the
proposition and let $E_0=[0,1]$. Define $E_{k}$ as the union of
all the intervals of $H_{n_{k}}$ contained in $E_{k-1}$. Then
$E_k$ is built up of intervals of length at least
$\frac{1}{\g(2n_k)}$ and separated by at least
$\e_k=\frac{1}{8n_k^2}$. Moreover, since
$\frac{1}{\g(2n_{k-1})}\ge \frac{3}{n_k}$, each interval of
$E_{k-1}$ contains at least
\[
m_k:=\frac{n_k^2}{6\log(n_k)\g(2n_{k-1})}
\]
intervals of $E_k$.

Now we can apply \prettyref{lem:cantorgeneral} to $E=\bigcap E_k$. Recall that $\e_k$ denotes the separation between $k$-level intervals and $m_k$ denotes the number of children of each of them. Consider $h\prec\h$. Then
\begin{eqnarray*}
 D^h_k & = & m_1\cdot m_2\cdots m_{k-1}h(\e_km_k)\\
    & = & \frac{6^{-(k-2)}n^2_2\cdots n^2_{k-1}}{\log(n_2)\cdots\log(n_{k-1})\g(n_1)\cdots\g(n_{k-2})}h\left(\frac{6}{\log(n_k)\g(n_{k-1})}\right).\\
\end{eqnarray*}
Now we note that $n_k\ge \log(n_k)$ and, by hypothesis $(A)$, we
also have that  $n_k\ge\g(2n_{k-1})\ge \g(n_{k-1})$. In addition,
$h$ is doubling, therefore it follows that we can bound the first
factor to obtain that
\begin{eqnarray*}
    D^h_k & \gtrsim &\frac{6^{-k}n^2_{k-1}}{\log(n_{k-1})\g(n_{k-2})}\frac{1}{\Delta(h,\h)\left(\frac{1}{\log(n_k)\g(n_{k-1})}\right)\left(\g^{-1}\left(\log(n_k)\g(n_{k-1})\right)\right)^2}\\
    & \gtrsim &\frac{6^{-k}n^2_{k-1}}{\g^{2}(n_{k-2})}
    \frac{1}{\Delta(h,\h)\left(\frac{1}{\log(n_k)\g(n_{k-1})}\right){\left(\g^{-1}(\log(n_k))+n_{k-1}\right)^2}}\\
\end{eqnarray*}
since, by hypothesis $(B)$, $n_k$ satisfies $\log(n_{k-1})\le
\g(n_{k-2})$ and $\g$ satisfies \prettyref{eq:logg-1}. Now, again
by hypothesis $(B)$,
\begin{equation*}
D^h_k \ge
\frac{1}{6^k\g^{2}(n_{k-2})}\frac{1}{\Delta(h,\h)\left(\frac{1}{\log(n_{k})\g(n_{k-1})}\right)}.
\end{equation*}
Thus, if
\[
\varliminf_k\frac{1}{6^k\g^{2}(n_{k-2})}\frac{1}{\Delta(h,\h)\left(\frac{1}{\log(n_{k})\g(n_{k-1})}\right)}
>0,
\]
then $\HH{h}(E)>0$ and therefore $\HH{h}(B_\g)>0$.
\end{proof}

We also have an example to illustrate this last result.
\begin{example}\label{ex:loglow}
Define $\g_r(x)=e^{x^\frac{2}{r}}$ for $r>0$ and consider the set $B_{\g_r}$.  Then $\h_r(x)=\frac{1}{\log^r(\frac{1}{x})}$ will be an expected lower bound for the dimension function for the set $B_{\g_r}$. Consider the chain $h_\theta(x)=\frac{1}{\log^\theta(\frac{1}{x})}$ ($0<\theta<r$), which satisfy $h_\theta\prec \h_r$. In this context, $\Delta(h_\theta,\h_r)(x)=\log^{\theta-r}(\frac{1}{x})$. Define the sequence $n_k$ as follows:
\[
    n_k=e^{kn^\frac{2}{r}_{k-1}}.
\]
Clearly the sequence is admissible, since
\begin{itemize}
\item[(A)] $n_{k}\ge 3\g(2n_{k-1})$, and
\item[(B)] $\log(n_k)\le \g(n_{k-1})$.
\end{itemize}
Inequality \prettyref{eq:critlow} now becomes
\begin{equation*}
D_k^h  \gtrsim
\frac{(\log\log(n_k)+n^\frac{2}{r}_{k-1})^{r-\theta}}{6^ke^{2n^\frac{2}{r}_{k-2}}}
\ge
\frac{n_{k-1}^{2\frac{r-\theta}{r}}}{6^ke^{2n^\frac{2}{r}_{k-2}}}.
\end{equation*}
Finally, for any $\e>0$ and  $M>0$, $n_k$ satisfies, for large
$k$,
\[
    \frac{n^\e_{k-1}}{6^ke^{Mn^\frac{2}{r}_{k-1}}}=
    \frac{e^{\e{kn_{k-1}^\frac{2}{r}}}}{6^ke^{Mn_{k-1}^\frac{2}{r}}}
    =\frac{e^{(\e k -M)n_{k-1}^\frac{2}{r}}}{6^k}\ge 1,
\]
so we conclude that $\varliminf_k D_k^h>0$. Therefore the set $E\sub B_{\g_r}$ constructed in the proof above satisfies $\HH{h_\theta}(E)>0$ for all $\theta<r$.
\end{example}

\subsection{Another Jarn\'ik type Theorem and proof of \prettyref{lem:fiber}}

For the proof of \prettyref{lem:fiber} we will need a different but essentially equivalent formulation of Jarn\'ik's theorem. We first recall that in the proof of \prettyref{thm:logFalpha} we have used the following theorem (see for example \cite{fal86})
\begin{theorem}\label{thm:jarnikA}
 Let $\n=\{n_j\}_j$ be a increasing sequence with $n_{j+1}\ge n_j^j$ for all $j\in\N$. For $0<\alpha\le1$, if $A^\n_\alpha$ is defined as
\[
A^\n_\alpha=\left\{x\in[0,1]\setminus \Q: \forall j\ \exists\
p,q\ ; q\le n_j^\alpha; |x-\frac{p}{q}|<\frac{1}{n_j^2}\right\},
\]
then $\dim_H(A^\n_\alpha)=\alpha$.
\end{theorem}\index{Aa@$A^\n_\alpha$}

For the proofs of \prettyref{thm:jarnikA} and
\prettyref{thm:jarnikB}, we refer the reader to \cite{jar31},
\cite{bes34}, \cite{egg52}, \cite{fal86}, and \cite{fal03}.

Now we want to relate the sets $A_\alpha$ and $B_\beta$ and their generalized versions. It is clear that for any $\alpha\in (0,1]$, we have the inclusion
$A_\alpha\subset B_\frac{2}{\alpha}$. For $\alpha\in (0,1]$, if $x\in A_\alpha$ then for each $j\in\N$ there exists a rational $\frac{p_j}{q_j}$ with $q_j\le n^\alpha_j$ such that $|x-\frac{p_j}{q_j}|<n_j^{-2}$, which is equivalent to $|xq_j-p_j|<q_jn_j^{-2}$. Therefore $|xq_j-p_j|\le q_j^{1-\frac{2}{\alpha}}$. Observe that if there were only finite values of $q$  for a given $x$, then $x$ has to be rational. For if $q_j=q_{j_0}$ for all $j\ge j_0$, then $|x-\frac{p_j}{q_{j_0}}|\to 0$ and this implies that $x\in \Q$. We conclude then that, for any $x\in A_\alpha$, $\|xq\|<\frac{1}{q^{\frac{2}{\alpha}-1}}$ for infinite many $q$ and therefore $x\in B_{\frac{2}{\alpha}}$. However, since the dimension of $A^\n_\alpha$ coincides with the one of
$B_\frac{2}{\alpha}$, one can expect that both sets have approximately comparable sizes.

We introduce the following  definition, which is the extended version of the definition of the  set $A^\n_\alpha$ in \prettyref{thm:jarnikA}.

\begin{definition}\label{def:jarnikh}
Let $\n=\{n_j\}_j$ be any increasing nonnegative sequence of
integers. Let $\f$ be a increasing function  defined on $\R_+$.
Define the set
\begin{equation*}
A^\n_\f:=\left\{x\in[0,1]\setminus \Q: \forall j\ \exists\ p,q\ ;
q\le \f(n_j); |x-\frac{p}{q}|<\frac{1}{n_j^2}\right\}.
\end{equation*} \index{Af@$A^\n_\alpha$}
\end{definition}
The preceding observation about the inclusion $A_\alpha\subset
B_\beta$ can be extended to this general setting. For a given
$\g$ as in the definition of $B_\g$, define
$\Gamma_\g(x)=\g^{-1}(x^2)$. Then the same calculations show that
$A^\n_{\Gamma_\g} \subset B_\g$.

We will need a converse relation between those sets, since we
want to prove a lower bound for the sets $A^\n_\f$ from the
estimates provided in \prettyref{pro:Bglow}.

\begin{lemma}\label{lem:EksubAf}
Let $\g$ and $B_\g$ be as in \prettyref{pro:Bglow}. Define
$\Gamma_\g(x)=\g^{-1}(x^2)$. Then, if $\m=\{m_k\}$ is the
sequence defining the set $E$ in the proof of
\prettyref{pro:Bglow}, then the set $E$ is contained in
$A^\n_{\Gamma_\g}$, where $\n=\{n_k\}=\{\g(m_k)^\frac{1}{2}\}$.
\end{lemma}
\begin{proof}
 Recall that in the proof of \prettyref{pro:Bglow} we define the sets $G'_q$ as a union of intervals of the form $I=\left(\frac{r}{q}-\frac{1}{\g(q)};\frac{r}{q}+\frac{1}{\g(q)}\right)$. The sets $H_n$ were defined as $   H_n:=\bigcup_{p\in\mathcal{P}_n^{2n}}G'_p$, where $\mathcal{P}_n^{2n}$ is the set of primes between $n$ and $2n$. We can therefore write
 \begin{equation*}
   H_n:=\bigcup I^n_j.
 \end{equation*}
Now, given a sequence $\m=\{m_k\}$, for each $k$, the set $E_k$
is defined as the union of all the intervals of $H_{m_k}$ that
belong to $E_{k-1}$, where $E_0=[0,1]$. If $E=\bigcap E_k$, any
$x\in E$ is in $E_k$ and therefore in some of the $I_j^{m_k}$. It
follows that there exists integers $r$ and $q$, $q\le 2m_k$ such
that
\begin{equation*}
 \left|x-\frac{r}{q}\right|<\frac{1}{\g(q)}<\frac{1}{\g(m_k)}=\frac{1}{n_k^2},\qquad q\le 2\g^{-1}(n_k^2).
\end{equation*}
Therefore $E\subset A_{\Gamma_\g}^\n$.
\end{proof}

We remark that the above inclusion implies that any lower estimate on the size of $E$ would also be a lower estimate for
$A_{\Gamma_\g}^\n$.

We now conclude the proof of \prettyref{thm:sqrth3/2} by proving \prettyref{lem:fiber}:

\begin{proof}[Proof of \prettyref{lem:fiber}:]
Let $\h(x)=\frac{1}{\log(\frac{1}{x})}$.  For $r>1$, consider the function $\g_r$, the sequence $\m=\{m_k\}$ and the set $E_r$ as in \prettyref{ex:loglow}. Define $\f=\Gamma_{\g_r}$, $\n$ and  $A^\n_{\f}$ as in \prettyref{lem:EksubAf}. It follows that $\f(x)=\log(x^2)^\frac{r}{2}$, $n_j=e^{\frac{1}{2}n_{j-1}^{\frac{4}{r}j}}$ and
\begin{equation*}
A^\n_\f:=\left\{x\in[0,1]\setminus \Q: \forall j\ \exists\ p,q\ ;
q\le \f(n_j); |x-\frac{p}{q}|<\frac{1}{n_j^2}\right\}.
\end{equation*}
Note that \prettyref{lem:EksubAf} says that the inclusion $E\subset A^\n_{\Gamma_\g}$ \emph{always} holds, for any defining function $\g$, where $E$ is the \emph{substantial portion} of the set $B_\g$ (see \prettyref{pro:Bglow}). But we need the positivity of $\HH{\h}(E)$ to conclude that the set $A^\n_{\Gamma_\g}$ also has positive $\HH{\h}$-measure. For the precise choices of $\h$ and $\g$, we obtain this last property from \prettyref{ex:loglow}. Precisely, $\HH{\h}(E_r)>0$ and therefore the set $A^\n_\f$ has positive $\HH{\h}$-measure. This concludes the proof of \prettyref{lem:fiber} and therefore the set constructed in the proof of \prettyref{thm:sqrth3/2} fulfills the condition of being an $F_\h$-set.
\end{proof}

\section*{Acknowledgments}
We would like to thank to Pablo Shmerkin for many interesting
suggestions an remarks about this problem.


text:
\begin{document}
\title{Power-Optimal Feedback-Based Random Spectrum Access for an Energy Harvesting Cognitive User
\thanks{This work was made possible by NPRP grants 6-1326-2-532 and 4-1034-2-385 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.}
}

\author{\IEEEauthorblockN{Mahmoud Ashour$^\dag$, Ahmed El Shafie$^{*\ddag}$, Amr Mohamed$^\dag$ and Tamer Khattab$^\ddag$}

\IEEEauthorblockA{$^\dag$ Computer Science and Engineering Dept., Qatar University, Doha, Qatar.\\
$^*$ Wireless Intelligent Networks Center (WINC), Nile University, Giza, Egypt.\\
$^\ddag$ Electrical Engineering Dept., Qatar University, Doha, Qatar.\\
}
\{m.ashour, ahmed.elshafie, amrm, tkhattab\}@qu.edu.qa
}

\maketitle

\begin{abstract}
In this paper, we study and analyze cognitive radio networks in which secondary users (SUs) are equipped with Energy Harvesting (EH) capability. We design a random spectrum sensing and access protocol for the SU that exploits the primary link's feedback and requires less average sensing time. Unlike previous works proposed earlier in literature, we do not assume perfect feedback. Instead, we take into account the more practical possibilities of overhearing unreliable feedback signals and accommodate spectrum sensing errors. Moreover, we assume an interference-based channel model where the receivers are equipped with multi-packet reception (MPR) capability. Furthermore, we perform power allocation at the SU with the objective of maximizing the secondary throughput under constraints that maintain certain quality-of-service (QoS) measures for the primary user (PU).

\end{abstract}

\begin{keywords}
Energy harvesting, cognitive radio, feedback, quality-of-service, multi-packet.
\end{keywords}

\section{Introduction}

Cognitive radio is a promising technology that efficiently utilizes the radio spectrum \cite{zhao2007survey}. Its essence resides in allowing secondary users (SUs) to access the spectrum while the primary users (PUs) are silent. Moreover, secondary nodes can use the spectrum concurrently with primary nodes in condition that they guarantee certain quality-of-service measures for the primary network.

Cognitive radio networks may involve some battery-based nodes and hence, energy management becomes a very important aspect. In many applications, the SU is a battery-constrained device. Therefore, its operation, which includes spectrum sensing and access, depends largely on its energy resources. Recently, energy harvesting has been gaining increasing worldwide interest. It is an emerging technology that is considered a promising solution to wireless energy-constrained networks. The significance of this technology resides in its capability to prolong wireless networks lifetime \cite{c0}. With such technology, nodes are capable of collecting energy from the surrounding environment \cite{c1,survey}.

Studying data transmission by an energy harvester with a rechargeable battery has received a lot of attention in the literature \cite{lei2009generic,sharma2010optimal,ho2010optimal,yang2010transmission,yang2010optimal,tutuncuoglu2010optimum,pappas2011optimal,krikidis2012stability}. Using a dynamic programming framework, Lei \emph{et al.} \cite{lei2009generic} derived the optimal online policy for controlling admissions into the data buffer. In \cite{sharma2010optimal}, Sharma \emph{et al.}
investigate the energy management policies that stabilize the data queue of a single-user communication. The authors derive some delay-optimal properties. For energy harvesting systems in
a time-constrained slotted setting, the authors in \cite{ho2010optimal} discussed the throughput-optimal energy allocation. In \cite{yang2010transmission,yang2010optimal}, Yang \emph{et al.} minimize the transmission completion time of an energy harvesting system. The optimal solution is obtained using a geometric framework. In \cite{tutuncuoglu2010optimum}, the authors solve the problem of maximizing the amount of data transmitted in a finite time horizon. The network under consideration consists of energy harvesting transmitters with batteries of finite energy storage.

In the context of cognitive communications, the authors in \cite{pappas2011optimal} consider a simple network composed of one PU and one SU. The PU is assumed to be equipped with a rechargeable battery (energy queue with random arrivals). The energy queue is assumed to be modelled as a decoupled ${\rm M/D/1}$ queue with Bernoulli arrivals and unity service rate. The SU is plugged to a reliable power supply; it always has energy/power. The maximum stable throughput region was derived using the dominant system approach \cite{dominance_approach}.

The optimal sensing and access policies for an energy harvesting SU based on Markov decision processes (MDPs) have been investigated in \cite{Sultan}. In \cite{ElSh1312:Optimal}, the optimal sensing duration of an energy harvesting SU is selected randomly at the beginning of every time slot from a predetermined set. The stable throughput is obtained via optimizing over the possibility of choosing certain sensing durations.

Due to the broadcast nature of wireless channels, primary feedback can be overheard by all nodes in the network. The main problem with schemes that employ spectrum sensing only is that sensing does not inform the SU about the impact of its transmissions on the primary receiver. This issue induced interest in leveraging the feedback sent by the primary receiver at the end of time slots to the primary transmitter in order to optimize the secondary transmission strategies. In \cite{eswaran2007bits}, Eswaran \emph {et al.} assume that the SU observes the feedback from the primary
receiver as it reflects the achieved primary rate. The SU aims at maximizing its throughput while guaranteeing a certain primary packet rate. The authors of \cite{lapiccirella2010cognitive} use a partially observable Markov decision process (POMDP) to optimize the
secondary action using the spectrum sensing outcome and primary Automatic Repeat reQuest (ARQ) feedback.\cite{huang2010distributed} investigates the secondary power control based on primary feedback. In \cite{wimob} and \cite{ourletter}, El Shafie \emph{et al.} derive the maximum stable throughput of a rechargeable SU sharing the channel with a PU. The SU perfectly overhears primary feedback.

In this paper, we analyze the performance of an energy harvesting SU sharing the spectrum with a PU. We propose a new random spectrum sensing and access scheme based on the outcome of the primary feedback signals overheard by the SU as well as the SU's spectrum sensing quality.
Unlike previous works proposed earlier in literature, we investigate the impact of erroneous primary feedback signals on the performance of both the PU and SU.

Our contributions in this paper can be summarized as follows. We assume feedback errors at the SU while receiving the primary feedback signals. In contrast to most of the literature, e.g., \cite{eswaran2007bits,lapiccirella2010cognitive,huang2010distributed,wimob,ourletter}, we investigate the impact of feedback errors on the secondary throughput as well as the average primary packet delay. Moreover, in contrast to \cite{wimob}, we consider Poisson energy arrivals at the energy queue and perform power allocation at the SU to achieve the maximum throughput. We propose a new spectrum sensing and access based on the joint outcome of primary feedback reception at the SU and the spectrum sensing process. We optimize over the sensing and access probabilities such that the secondary throughput is maximized, subject to the constraint that the primary queue is kept stable. We then study the worst case scenario for the PU and append a queueing delay constraint to the optimization problem that carries further quality-of-service (QoS) guarantees for the PU.

The rest of this paper is organized as follows. Section \ref{system_model} presents the system model along with the proposed spectrum access protocol. In Section \ref{performance_analysis}, we present the queueing analysis of both PU and SU and derive a closed-form expression for the average primary packet delay. Next, we derive lower and upper bounds on the secondary throughput in Section \ref{lower_and_upper_bounds}. Section \ref{numerical_results} shows our numerical results. Finally, concluding remarks are drawn in Section \ref{conclusion}.

\section{System Model}\label{system_model}
We consider the cognitive radio system shown in Fig. \ref{Fig1}.
The system consists of a PU ($\rm p$) transmitting its packets to a primary destination ($\rm d_{p}$). In addition, there exists an SU (${\rm s}$) communicating with a secondary destination ($\rm d_{s}$). The PU is plugged to a reliable power supply. This means that it never lacks energy/power. However, the SU is equipped with a rechargeable battery. We assume that the SU harvests energy from the surrounding environmental energy sources, e.g., solar energy, wind energy, etc. All nodes are equipped with infinite capacity buffers to store fixed-length ($\beta$ bits) data packets. Time is slotted and the duration of one time slot is $T$ seconds. The transmission of a packet takes one time slot.

The channel between every transmitter-receiver pair exhibits frequency-flat Rayleigh block fading, i.e., the channel coefficient remains constant for one time slot and changes independently from a slot to another. Moreover, a zero-mean additive white Gaussian noise of power spectral density $\mathcal{N}_{\circ}$ Watts/Hz is taken into account. The channel coefficient between transmitter $\rm i$ and receiver $\rm j$ at the $n$th time slot is denoted by $h_{\rm i,j}[n]$. According to the Rayleigh fading assumption, $h_{\rm  i,j}[n]$ is a complex Gaussian zero-mean random variable with variance $ \sigma_{\rm i,j}^{2}$, i.e., $h_{\rm i,j}[n] \sim \mathcal{CN}(0,\sigma_{\rm i,j}^{2})$. Therefore, $|h_{\rm i,j}[n]|^2$ is exponentially-distributed with rate $1/\sigma_{\rm i,j}^{2}$, i.e., $|h_{\rm i,j}[n]|^2 \sim \text{exp}(1/\sigma_{\rm i,j}^{2})$. All links are considered statistically independent.

Instead of assuming a simple packet-erasure model, we consider simultaneous multi-packet transmission capability. This opens room for both the PU and SU to use the spectrum concurrently. Assuming that the receivers are equipped with multi-packet reception (MPR) capability, transmitted data packets can survive the interference caused by concurrent transmissions if the received signal-to-interference-and-noise ratio (SINR) exceeds the threshold required for successful decoding. The SU performs probabilistic spectrum sensing. We take into account erroneous spectrum sensing outcomes.

Next, we present the queueing model of the system followed by the description of the proposed spectrum access protocol.

\subsection{Queueing Model}
The queues involved in system analysis, shown in Fig. \ref{Fig1}, are described as follows:
\begin{itemize}
\item $Q_{\rm p}$: stores the data packets of the PU.

\item $Q_{\rm s}$: stores the data packets of the SU.

\item $Q_{\rm e}$: stores the harvested energy at the SU.
\end{itemize}

The bursty nature of information sources is taken into account through modelling the data arrivals at the PU as a Bernoulli process with rate $\lambda_{\rm p}$ (packets/slot). In other words, at any given time slot, a packet arrives at the PU with probability $\lambda_{\rm p}<1$. On the other hand, we assume that $Q_{\rm s}$ is backlogged, i.e., the SU always has packets awaiting transmission to $\rm d_s$. The arrivals  at the energy queue are assumed to follow a stationary Poisson process with rate $\lambda_{\rm e}$. This captures the random availability of ambient energy sources. The arrival processes at $Q_{\rm p}$ and $Q_{\rm e}$ are independent of each other, and are independent and identically-distributed (i.i.d) across time slots. Upon successful reception of a primary packet, $\rm  d_{p}$ broadcasts an acknowledgement (ACK). However, if $\rm d_{p}$ fails to decode a primary packet, it broadcasts a negative-acknowledgement (NACK). ACKs and NACKs are assumed instantaneous and can be heard by the PU and SU. The PU receives the feedback sent by $\rm d_{p}$ reliably with probability $1$. On the contrary, the SU overhears reliable primary feedback with probability $q$.

The instantaneous evolution of queue lengths is captured as
\begin{align}\label{queue evolution}
Q_{\rm i}[n+1]= \left(Q_{\rm i}[n] - L_{\rm i}[n]\right)^{+} + \mathcal{A}_{\rm i}[n], ~\rm i \in \{\rm p,e\}
\end{align}
where $(z)^{+}=\text{max}(z,0)$ and $Q_{\rm i}[n]$ denotes the number of packets in the $i$th queue at the beginning of the $n$th time slot. $L_{\rm i}[n]$ and $\mathcal{A}_{\rm i}[n]$ denote the departures and arrivals corresponding to the $\rm i$th queue in the $n$th time slot, respectively.

\begin{figure}[t]
\begin{center}
\includegraphics[width=1\columnwidth , height=0.67\columnwidth]{system_model.eps}
\caption{Cognitive radio network under consideration.} \label{Fig1}
\end{center}
\vspace{-5mm}
\end{figure}

\subsection{Spectrum Access Protocol}\label{system_model_B}
Whenever $Q_{\rm p}$ is non-empty, the PU transmits a packet with average power $P_{\rm p}$. However, the SU probably attempts to transmit the head-of-line packet in its data queue if its battery has got the energy required to support that transmission. The spectrum access protocol employed at the SU exploits the available primary feedback signals. The SU may overhear nothing if there is no primary transmission. However, it may overhear an ACK or a NACK if the primary receiver correctly decodes, or fails to decode, the primary transmission, respectively, while the SU decodes the feedback signals correctly. Otherwise, the SU may overhear un-decodable feedback signals. Given that a primary transmission takes place in a given time slot, the SU is capable of decoding the primary feedback signals with probability $q$ as indicated earlier. If the SU receives nothing, an ACK or un-decodable feedback, it gains no information about the PU's activity in the next time slot. Therefore, the SU operates as follows.
\begin{itemize}
\item If the energy required for transmission is available at $Q_{\rm e}$, it performs spectrum sensing with probability $\alpha_{\rm s}$ from the beginning of the time slot for a duration $\tau < T$ seconds to detect the possible activity of the PU.

\item The SU transmits with probability $\alpha_{\rm f}$ or $\alpha_{\rm b}$ if the PU is detected to be idle or busy, respectively.

\item At the beginning of the time slot, if the SU decides not to sense the spectrum (which happens with probability $1-\alpha_{\rm s}$), it immediately decides whether to transmit with probability $\alpha_{\rm t}$ or to remain idle for the rest of the time slot with probability $1-\alpha_{\rm t}$.
\end{itemize}
Whenever the SU transmits in these cases, it uses an average power $P_{\rm s}^{\left(1\right)}$ when the PU is sensed to be inactive; and $P_{\rm s}^{\left(2\right)}$ when the PU is sensed to be active. Note that $P_{\rm s}^{\left(2\right)}$ should be lower than $P_{\rm s}^{\left(1\right)}$ to reduce the expected interference on the primary transmission. On the other hand, if a NACK is overheard by the SU, it knows that the PU will retransmit the lost packet during the next time slot with probability $1$. Being sure that the PU is active in the next slot, the SU does not need to perform spectrum sensing. Therefore, it accesses the channel from the beginning of the time slot with probability $\alpha_{\rm r}$. In that case, the SU transmits with average power $P_{\rm s}^{\left(3\right)} ~\le~ P_{\rm s}^{\left(2\right)}~\le~  P_{\rm s}^{\left(1\right)}$ to reduce its interference to the primary receiver.

We assume that energy dissipation from $Q_{\rm e}$ happens when the SU transmits data. The energy consumed in spectrum sensing and primary feedback decoding is negligible and out of the scope of this paper.

\begin{figure}[t]
\begin{center}
\includegraphics[width=1\columnwidth , height=0.67\columnwidth]{MC2.eps}
\caption{Markov chain of the PU's queue-length evolution.} \label{Fig2}
\end{center}
\vspace{-5mm}
\end{figure}

\section{Performance Analysis}\label{performance_analysis}
In this section, we provide the queueing analysis of the proposed access scheme. In particular, we derive stability conditions on the queues involved in the system. The system is considered stable when all of its queues are stable. Queue stability is loosely defined as having a bounded queue size, i.e., the number of packets in the queue does not grow to infinity as time progresses \cite{sadek}. Furthermore, we provide a closed-form expression for the average primary packet delay.

\subsection{Queueing Analysis}

The PU's queue-length evolution Markov chain (MC) is shown in Fig. \ref{Fig2}\footnote{State-self transitions are not depicted on the graph for visual clarity. Throughout the paper,
$\overline{y}=1-y$.}. We refer to the probability of the queue having $\rm k$ packets and transmitting for the first time as $\pi_{\rm k}$, where $F$ in Fig. \ref{Fig2} denotes the first transmission. The probability of the queue having $\rm k$ packets and retransmitting is $\chi_{\rm k}$, where $R$ in Fig. \ref{Fig2} denotes retransmission. The probability of successful transmission of a PU's packet in case of first transmission and a retransmission is given by $\Omega_{\rm p}$ and $\Gamma_{\rm p}$, respectively. We proceed with calculating these probabilities.
\begin{equation}\label{alpha_p}
\Omega_{\rm p}=\mathbb{P} \gamma+\overline{\mathbb{P}}\mathbf{P}_{\rm 0}(P_{\rm p},0)
\end{equation}
where $\mathbb{P}$ is the probability that $Q_{\rm e}$ has got the amount of energy sufficient for one secondary packet transmission, $\mathbf{P}_{\rm i}(P_{\rm A},P_{\rm B})$ is derived in the Appendix and $\gamma$ is given by
\begin{align}\label{gamma}
\gamma  &= \overline{\alpha_{\rm s}}\bigg\{\alpha_{\rm t}\mathbf{P}_{\rm 0}(P_{\rm p},P_{\rm s}^{\left(1\right)}) + \overline{\alpha_{\rm t}}\mathbf{P}_{\rm 0}(P_{\rm p},0)\bigg\}\notag \\
&+\alpha_{\rm s}{\rm P}_{\rm MD}\bigg\{\alpha_{\rm f}\mathbf{P}_{\rm 0}(P_{\rm p},P_{\rm s}^{\left(1\right)})+ \overline{\alpha_{\rm f}}\mathbf{P}_{\rm 0}(P_{\rm p},0)\bigg\} \notag \\
&+\alpha_{\rm s}\overline{{\rm P}_{\rm MD}}\bigg\{\alpha_{\rm b}\mathbf{P}_{\rm 0}(P_{\rm p},P_{\rm s}^{\left(2\right)}) + \overline{\alpha_{\rm b}}\mathbf{P}_{\rm 0}(P_{\rm p},0)\bigg\}.
\end{align}
We denote the probability of miss-detection by ${\rm P}_{\rm MD}$. On the other hand, it can be shown that $\Gamma_{\rm p}$ is given by
\begin{table}
\renewcommand{\arraystretch}{2}
\begin{center}
\begin{tabular}{ |c |c|  }
    \hline
    $\eta$ & $\lambda_{\rm p}\Omega_{\rm p}+\left(1-\lambda_{\rm p}\right)\Gamma_{\rm p}$ \\[5pt]\hline
  $\pi_{\rm \circ}$ & $\frac{\eta-\lambda_{\rm p}}{\Gamma_{\rm p}}$ \\[5pt]\hline
  $\chi_{\rm \circ}$ & 0 \\[5pt]\hline
  $\pi_1$& $\pi_{\rm \circ}\tfrac{\lambda_{\rm p}}{1-\lambda_{\rm p}}\frac{\lambda_{\rm p}+\left(1-\lambda_{\rm p}\right)\Gamma_{\rm p}}{\eta}$  \\[5pt]\hline
  $\chi_1$ & $\pi_{\rm \circ}\frac{\lambda_{\rm p}}{\eta}\left(1-\Omega_{\rm p}\right)$ \\ [5pt]\hline
  $\pi_{\rm k},{\rm k}\geq 2$ & $\pi_{\rm \circ}\frac{\lambda_{\rm p}\left(1-\Omega_{\rm p}\right)}{\left(1-\eta\right)^2} \bigg[ \frac{\lambda_{\rm p} \left(1-\eta\right)}{\left(1-\lambda_{\rm p}\right) \eta}  \bigg]^{\rm k}$ \\[5pt]\hline
  $\chi_{\rm k},{\rm k}\geq 2$ & $\pi_{\rm \circ}\frac{\left(1-\lambda_{\rm p}\right)\left(1-\Omega_{\rm p}\right)}{\left(1-\eta\right)^2} \bigg[ \frac{\lambda_{\rm p} \left(1-\eta\right)}{\left(1-\lambda_{\rm p}\right) \eta}  \bigg]^{\rm k}$ \\[5pt]\hline
  $\sum_{\rm k=1}^{\infty}\pi_{\rm k}$ & $\pi_{\rm \circ}\frac{\lambda_{\rm p}\Gamma_{\rm p}}{\eta-\lambda_{\rm p}}=\lambda_{\rm p}$\\[5pt]\hline
   $\sum_{\rm k=1}^{\infty}\chi_{\rm k}$ & $\pi_{\rm \circ}\frac{\lambda_{\rm p}}{\eta-\lambda_{\rm p}}\left(1-\Omega_{\rm p}\right)=\frac{\lambda_{\rm p} }{\Gamma_{\rm p}}\left(1-\Omega_{\rm p}\right)$
  \\[5pt]\hline
\end{tabular}
\vspace{+4mm}
\caption{Steady-state distribution for the PU's MC.}
\label{table}
\end{center}
\vspace{-5mm}
\end{table}
\begin{equation}\label{Gamma_p}
\Gamma_{\rm \!p}\!=\!\mathbb{P}\bigg[q\bigg\{\!\alpha_{\rm r}\mathbf{P}_{\rm 0}(P_{\rm p},P_{\rm s}^{\left(3\right)})
\!+\!\overline{\alpha_{\rm r}}\mathbf{P}_{\rm 0}(P_{\rm p},0)\!\bigg\}\!+\!\overline{q}\gamma
\bigg]\!+\!\overline{\mathbb{P}}\mathbf{P}_{\rm 0}(P_{\rm p},0).
\end{equation}
We solve for the steady-state distribution of the PU's MC.
Solving the state balance equations of the MC depicted in Fig. \ref{Fig2}, we obtain the state probabilities which are provided in Table \ref{table}. The probability $\pi_0$ is obtained using the normalization condition $\sum_{\rm k=0}^{\infty} (\pi_{\rm k}+\chi_{\rm k})=1$.
It should be noticed that $\lambda_{\rm p} < \eta$, where $\eta$ is defined in Table \ref{table}, is a condition for the sum $\sum_{\rm k=0}^{\infty} (\pi_{\rm k}+\chi_{\rm k})$ to exist. This condition ensures the existence of a stationary distribution for the MC. Furthermore, it guarantees the stability of $Q_{\rm p}$, i.e., $Q_{\rm p}$ has a non-zero probability of being empty ($\pi_{\rm 0}>0$).

From (\ref{alpha_p}) and (\ref{Gamma_p}), we notice that $\Omega_{\rm p}$ and $\Gamma_{\rm p}$ depend on the state of $Q_{\rm e}$ through the term $\mathbb{P}$. Therefore, we need to model the energy dissipation from the battery of the SU, i.e., $Q_{\rm e}$. Let ${\rm P}_{\rm FA}$ denote the probability of false alarm. According to the proposed access protocol described in Section \ref{system_model_B}, the mean service rate of $Q_{\rm e}$ is given in (\ref{mu_e}) at the top of the following page.

                \begin{figure*}[!t]
\normalsize
\setcounter{equation}{4}
\begin{align}\label{mu_e}
\mu_{\rm e}\!&=\!\pi_{\rm 0}\bigg\{\overline{\alpha_{\rm s}} \alpha_{\rm t} P_{\rm s}^{\left(1\right)} T\! +\! \bigg[\alpha_{\rm s}\alpha_{\rm f}\overline{{\rm P}_{\rm FA}}
P_{\rm s}^{\left(1\right)}+\alpha_{\rm s}\alpha_{\rm b}{\rm P}_{\rm FA} P_{\rm s}^{\left(2\right)}\bigg]T_{\rm s}\bigg\}\!+\!
\left(\displaystyle\sum_{\rm k=1}^{\infty}\!\pi_{\rm k}\right)\!\!\bigg\{ \overline{\alpha_{\rm s}}\alpha_{\rm t}P_{\rm s}^{\left(1\right)}T
\!+\!
\alpha_{\rm s}\bigg[\alpha_{\rm f}{\rm P}_{\rm MD} P_{\rm s}^{\left(1\right)}\!+\!\alpha_{\rm b}\overline{{\rm P}_{\rm MD}}P_{\rm s}^{\left(2\right)}\bigg]T_{\rm s} \bigg\} \notag \\ &\,\,\,\,\,\,\,\,\
 +\left(\!\displaystyle\sum_{\rm k=1}^{\infty}\!\chi_{\rm k}\!\!\right)\!
\bigg\{\!\!q\alpha_{\rm r}P_{\rm s}^{\left(3\right)}T\!\!+\!\!\overline{q} \!\bigg(\! \overline{\alpha_{\rm s}}\alpha_{\rm t} P_{\rm s}^{\left(1\right)} T \!+\!
\alpha_{\rm s}\bigg[\!\alpha_{\rm f}{\rm P}_{\rm MD} P_{\rm s}^{\left(1\right)}\!+\!\alpha_{\rm b}\overline{{\rm P}_{\rm MD}}P_{\rm s}^{\left(2\right)}\!\bigg]\!T_{\rm s}\!\!\bigg)\!\!\bigg\}, \ \text{where} \ T_{\rm s}=T-\tau.
\\
\label{mu_s}
\mu_{\rm s}\! &=\!\mathbb{P}\bigg[ \pi_{\rm 0}\bigg\{\overline{\alpha_{\rm s}} \alpha_{\rm t} \mathbf{P}_{\rm \!0}(P_{\rm s}^{\left(1\right)},0)\! +\! \alpha_{\rm s}\big[\alpha_{\rm f}\overline{{\rm P}_{\rm FA}}
\mathbf{P}_{\rm \!1}(P_{\rm s}^{\left(1\right)},0)+\alpha_{\rm b}{\rm P}_{\rm FA}\mathbf{P}_{\rm \!1}(P_{\rm s}^{\left(3\right)},0) \big]\!\bigg\} \notag \\
&\,\,\,\,\,\,\,\,\  +\!
\left(\!\displaystyle\sum_{\rm k=1}^{\infty}\!\!\pi_{\rm k}\!\right)\!\!\!\bigg\{\! \overline{\alpha_{\rm s}}\alpha_{\rm t}
\mathbf{P}_{\rm \!0}\!(P_{\rm s}^{\left(1\right)},P_{\rm p})
\!\!+\!\!
\alpha_{\rm s}\big[\alpha_{\rm f}{\rm P}_{\rm MD} \mathbf{P}_{\rm \!1}\!(P_{\rm s}^{\left(1\right)},P_{\rm p})+\alpha_{\rm b}\overline{{\rm P}_{\rm MD}}\mathbf{P}_{\rm \!1}\!(P_{\rm s}^{\left(3\right)},P_{\rm p})\big]\! \bigg\}
\notag \\ &\,\,\,\,\,\,\,\,\
+\!\left(\displaystyle\sum_{\rm k=1}^{\infty}\chi_{\rm k}\right)
\bigg\{ q \alpha_{\rm r}\mathbf{P}_{\rm 0}(P_{\rm s}^{\left(2\right)},P_{\rm p})
+\overline{q}\bigg(\overline{\alpha_{\rm s}}\alpha_{\rm t}\mathbf{P}_{\rm 0}(P_{\rm s}^{\left(1\right)},P_{\rm p})+
\alpha_{\rm s}\big[\alpha_{\rm f}{\rm P}_{\rm MD} \mathbf{P}_{\rm 1}(P_{\rm s}^{\left(1\right)},P_{\rm p})+\alpha_{\rm b}\overline{{\rm P}_{\rm MD}} \mathbf{P}_{\rm 1}(P_{\rm s}^{\left(3\right)},P_{\rm p})\big]
\!\bigg)\!\bigg\}\bigg]\!.
\end{align}
\hrulefill

\end{figure*}
It is obvious that the service rate of $Q_{\rm e}$ depends on the state of $Q_{\rm p}$, i.e., $\pi_{\rm k}$ and $\chi_{\rm k}$. Thus, $Q_{\rm p}$ and $Q_{\rm e}$ are two interacting queues. The relaxation of this interaction and the computation of $\mathbb{P}$ is provided in the next section.

We proceed next with characterizing the SU throughput, i.e., the mean service rate of $Q_{\rm s}$. It is given in (\ref{mu_s}) at the top of the following page.

The dependence of the SU throughput on both $Q_{\rm e}$ and $Q_{\rm p}$ is highlighted in (\ref{mu_s}). The SU transmits only when the energy required to support its transmission is available in $Q_{\rm e}$. This explains the role of $\mathbb{P}$ in (\ref{mu_s}). In addition, the SU's behavior depends on the state of the PU, i.e., the PU being in a first transmission or a retransmission state. By behavior here we refer to the decisions made by the SU concerning spectrum sensing and access and the choice of transmission powers.

\subsection{Average Primary Packet Delay}
Applying Little's law \cite{Bertsekas}, we obtain a closed-form expression for the average primary packet delay, which is given by
\begin{equation}
D_{\rm p}=\frac{1}{\lambda_{\rm p}}\sum_{\rm k=1}^{\infty} {\rm k} \left(\pi_{\rm k}+\chi_{\rm k}\right).
\end{equation}
\noindent Using the state probabilities provided in Table \ref{table},
\begin{equation}\label{delay_p}
\begin{split}
D_{\rm p} &=\frac{(\Omega_{\rm p}-\eta)(\eta-\lambda_{\rm p})^2+\left(1-\lambda_{\rm p}\right)^2 \left(1-\Omega_{\rm p}\right) \eta}{(\eta-\lambda_{\rm p})\left(1-\lambda_{\rm p}\right) \left(1-\eta\right) \Gamma_{\rm p}}.
  \end{split}
\end{equation}

\section{Bounds on Secondary Throughput and Problem Formulation}\label{lower_and_upper_bounds}
In this section, we derive lower and upper bounds on the throughput of the SU.
Our main goal now is to relax the interaction between $Q_{\rm e}$ and $Q_{\rm p}$ to be able to compute $\mathbb{P}$ and $\{\pi_{\rm k},\chi_{\rm k}\}_{\rm k=0}^{\infty}$ and hence, the throughput of the SU can be characterized.
\subsection{Lower Bound on $\mu_{\rm s}$}\label{lb_on_mu_s}
We analyze the effect of the states of $Q_{\rm p}$ and $Q_{\rm e}$ on the SU throughput. We begin first with $Q_{\rm p}$. If the PU's service rate is decreased, less number of packets probably depart $Q_{\rm p}$. This implies that the probability of $Q_{\rm p}$ being empty is lowered. Therefore, the PU is more likely to be active and hence, the interference on the SU is increased which lowers its throughput. This motivates us to derive lower bounds on $\Omega_{\rm p}$ and $\Gamma_{\rm p}$ provided in (\ref{alpha_p}) and (\ref{Gamma_p}), respectively. We note that the worst case scenario with respect to the PU occurs when it experiences continuous possible interference from the SU. Since the SU is assumed backlogged, continuous SU transmission possibly occurs when it always has the required energy, i.e., $\mathbb{P}=1$. Substituting by $\mathbb{P}=1$ in (\ref{alpha_p}) and (\ref{Gamma_p}),
\begin{eqnarray}
\Omega_{\rm p} &\geq & \gamma \label{alpha_p_lb} \\
\Gamma_{\rm p} &\geq & q\bigg\{\alpha_{\rm r}\mathbf{P}_{\rm 0}(P_{\rm p},P_{\rm s}^{\left(3\right)})
+\overline{\alpha_{\rm r}}\mathbf{P}_{\rm 0}(P_{\rm p},0)\bigg\}+\overline{q}\gamma \label{Gamma_p_lb}
\end{eqnarray}
where $\gamma$ is given by (\ref{gamma}).

On the other hand, we note from (\ref{mu_e}) that the energy dissipation from $Q_{\rm e}$, whenever a secondary transmission occurs, is one of four levels: (i) $P_{\rm s}^{\left(1\right)} T$, (ii) $P_{\rm s}^{\left(1\right)}T_{\rm s}$, (ii) $P_{\rm s}^{\left(2\right)}T_{\rm s}$ and (iii) $P_{\rm s}^{\left(3\right)}T$. We assume that $Q_{\rm e}$ dissipates $P_{\rm s}^{\left(1\right)} T$ for each secondary transmission provided that this energy is available in $Q_{\rm e}$. This provides an upper bound on the service rate of $Q_{\rm e}$ because $P_{\rm s}^{\left(1\right)} T$ is the one which has the highest level among the aforementioned four levels. Hence, the SU battery is depleted faster which negatively impacts its throughput. This assumption renders $Q_{\rm e}$ an $\text{M}/\text{D}/1$ queue with arrival rate $\lambda_{\rm e}$ and service rate $P_{\rm s}^{\left(1\right)} T$. Therefore, $\mathbb{P}$ is \cite{krikidis2012stability}
\begin{equation}\label{P_Qe_ub}
\mathbb{P}=\text{min}\left[1,\frac{\lambda_{\rm e}}{P_{\rm s}^{\left(1\right)} T}\right].
\end{equation}

We use the lower bounds on $\Omega_{\rm p}$ and $\Gamma_{\rm p}$ provided on the right hand sides of (\ref{alpha_p_lb}) and (\ref{Gamma_p_lb}), respectively, to compute the state probabilities of the PU's MC given by Table \ref{table}. Substituting by these state probabilities along with $\mathbb{P}$ as given by (\ref{P_Qe_ub}) in (\ref{mu_s}), we get a lower bound on the throughput of the SU.

\subsection{Upper Bound on $\mu_{\rm s}$}\label{ub_on_mu_s}
We use a similar approach to the one presented in Section \ref{lb_on_mu_s} to derive an upper bound on the SU throughput. When the service rate of the PU's queue increases, this increases the availability of time slots in which the PU is silent. Thus, the interference on the SU decreases and its throughput is enhanced. This motivates us to derive upper bounds on $\Omega_{\rm p}$ and $\Gamma_{\rm p}$.
The best case scenario with respect to the PU occurs when it experiences no interference from the SU, as if the SU does not exist. This happens when the battery of the SU is always empty, corresponding to $\mathbb{P}=0$. Substituting by $\mathbb{P}=0$ in (\ref{alpha_p}) and (\ref{Gamma_p}), we get
\begin{eqnarray}
\Omega_{\rm p} & \leq & \mathbf{P}_{\rm 0}(P_{\rm p},0) \label{alpha_p_ub} \\
\Gamma_{\rm p} & \leq & \mathbf{P}_{\rm 0}(P_{\rm p},0) \label{Gamma_p_ub}.
\end{eqnarray}

On the other hand, we assume that $Q_{\rm e}$ dissipates $\text{min}[P_{\rm s}^{\left(2\right)}T_{\rm s},P_{\rm s}^{\left(3\right)}T]$ for each secondary transmission provided that this energy is available in $Q_{\rm e}$. This provides a lower bound on the service rate of $Q_{\rm e}$. Hence, the SU battery lifetime increases which positively impacts its throughput. This assumption renders $Q_{\rm e}$ an $\text{M}/\text{D}/1$ queue with arrival rate $\lambda_{\rm e}$ and service rate $\text{min}[P_{\rm s}^{\left(2\right)}T_{\rm s},P_{\rm s}^{\left(3\right)}T]$. Therefore, $\mathbb{P}$ is given by \cite{krikidis2012stability}
\begin{equation}\label{P_Qe_lb}
\mathbb{P}=\text{min}\left[1,\frac{\lambda_{\rm e}}{\text{min}[P_{\rm s}^{\left(2\right)}T_{\rm s},P_{\rm s}^{\left(3\right)}T]}\right].
\end{equation}

We use the upper bounds on $\Omega_{\rm p}$ and $\Gamma_{\rm p}$ provided on the right hand sides of (\ref{alpha_p_ub}) and (\ref{Gamma_p_ub}), respectively, to compute the state probabilities of the PU's MC given by Table \ref{table}. Substituting by these state probabilities along with $\mathbb{P}$ as given by (\ref{P_Qe_lb}) in (\ref{mu_s}), we get an upper bound on the SU throughput.

\subsection{Problem Formulation}
Using the relevant mean service rates under each of the proposed bounds, for a fixed $\lambda_{\rm p}$, the following constrained optimization problem is solved numerically
\begin{align}\label{optimization}
& \underset{\alpha_{\rm s},\alpha_{\rm f},\alpha_{\rm t},\alpha_{\rm b},\alpha_{\rm r},P_{\rm s}^{\left(1\right)},P_{\rm s}^{\left(2\right)},P_{\rm s}^{\left(3\right)}}{\text{maximize}}
& & \mu_{\rm s} \notag \\
& ~~~~~ \text{subject to}
& & 0 \leq \alpha_{\rm s},\alpha_{\rm f},\alpha_{\rm t},\alpha_{\rm b},\alpha_{\rm r} \leq 1 \notag \\
&&& 0 \leq P_{\rm s}^{\left(1\right)},P_{\rm s}^{\left(2\right)},P_{\rm s}^{\left(3\right)} \leq P_{\rm \circ} \notag \\
&&& \lambda_{\rm p} < \eta \notag \\
&&& D_{\rm p} \leq D_{\rm \circ}.
\end{align}
We set maximum secondary transmission power constraint $P_{\rm \circ}$, and a threshold $D_{\rm \circ}$ below which the average primary packet delay is kept. Note that the optimization problem is solved at the secondary terminal. We use MatLab's fmincon to solve the optimization problems as in \cite{wimob,6568963,6177245,4472701,6648968} and the references therein.

\section{Numerical results}\label{numerical_results}
In this section, the performance of the proposed scheme is evaluated in terms of the SU throughput. We formulate and solve an optimization problem with the objective of maximizing the secondary throughput subject to certain guaranteed QoS measures for the PU. We solve for the optimal sensing and access probabilities as well as the average secondary transmission powers that maximize the SU throughput while simultaneously keeping the PU's queue stable and the primary packet delay below a certain threshold.

We show our results for packet-length $\beta=10$ bits, time slot duration $T=1$ second and sensing duration $\tau=0.3$ second. Sensing errors are taken into account through setting ${\rm P}_{\rm MD}={\rm P}_{\rm FA}=0.3$. All links are considered statistically equivalent where $\sigma_{\rm i,j}^{2}=1$ $\forall {\rm i} \in \{\rm p,s\}$ and ${\rm j} \in \{\rm d_{p}, d_{s}\}$. The bandwidth of these channels is set to $W=8$ Hz. The PU transmits its packets with average power $P_{\rm p}=20$ Watts. The power spectral density of noise is normalized to unity, i.e., $\mathcal{N}_{\rm \circ}=1$ Watts/Hz. We choose $P_{\rm \circ}=32$ Watts (corresponding to a signal-to-noise-ratio $=6$ dB if the SU is transmitting alone).
As indicated earlier, $\lambda_{\rm p}<\eta$ is the constraint that guarantees the stability of the PU's queue.

\begin{figure}[t]
\begin{center}
\includegraphics[width=1\columnwidth]{Le.eps}
\caption{Optimal lower and upper bounds on $\mu_{\rm s}$ versus $\lambda_{\rm p}$ for different values of $\lambda_{\rm e}$.} \label{Fig3}
\end{center}
\vspace{-5mm}
\end{figure}

\begin{figure}[t]
\begin{center}
\includegraphics[width=1\columnwidth]{q.eps}
\caption{Optimal lower bound on $\mu_{\rm s}$ versus $\lambda_{\rm p}$ for different values of $q$.} \label{Fig4}
\end{center}
\vspace{-5mm}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\columnwidth]{delay_vs_q.eps}
\caption{Optimal upper bound on $D_{\rm p}$ versus $q$ for different values of $\lambda_{\rm p}$.} \label{Fig5}
\end{center}
\vspace{-5mm}
\end{figure}

In Fig. \ref{Fig3}, we plot the resulting solution of (\ref{optimization}), with $\mu_{\rm s}$ given by the lower and upper bounds explained in Sections \ref{lb_on_mu_s} and \ref{ub_on_mu_s}, respectively, versus $\lambda_{\rm p}$ at a chosen value of $q=0.5$ and $D_{\rm \circ}=10$ time slots. This corresponds to a PU running a delay-tolerant application. In the figure's legend, we denote the upper bound and lower bound by `UB' and `LB', respectively. We investigate the effect of the energy arrival rate at $Q_{\rm e}$ on the SU throughput through plotting the results at different values of $\lambda_{\rm e}$. As we expected, the figure shows that the SU throughput is enhanced as $\lambda_{\rm e}$ increases. This is attributed to the fact that the restriction on the SU's activity originating from its energy resources is relaxed as $\lambda_{\rm e}$ increases. From the figure, we note that the gap between the inner and the outer bound decreases with the increase of $\lambda_{\rm e}$.

Next, we depict the effect of unreliable primary feedback signals on SU throughput and average primary packet delay. Towards this objective, we plot the optimal lower bound on $\mu_{\rm s}$ versus $\lambda_{\rm p}$ for different values of $q$ in Fig. \ref{Fig4}. We choose $\lambda_{\rm e}=20$ and $D_{\rm \circ}=10$. As shown in the figure, the throughput of the SU is enhanced as $q$ increases. This shows that the best case scenario occurs when the SU receives perfect primary feedback, i.e., $q=1$. However, the worst case scenario occurs when the SU overhears no reliable feedback from the PU, i.e., $q=0$. The case of $q=0$ boils down to the case in which the SU can not overhear/decode the primary feedback signals.
On the other hand, we evaluate the underlying average PU delay.
We use the resulting optimal sensing and access probabilities as well as average secondary transmission powers, used to plot Fig. \ref{Fig4}, in computing the average primary packet delay given by (\ref{delay_p}). It is worth noting that the primary delay computed through solving (\ref{optimization}) with the objective of maximizing the lower bound on $\mu_{\rm s}$ is an upper bound on the achievable PU delay. In Fig. \ref{Fig5}, we plot the PU's delay versus $q$ at selected values of $\lambda_{\rm p}$. Thus, it can be seen that the primary delay is a decreasing function of $q$. Therefore, we reach the conclusion that reliable feedback signals overheard by the SU is not only in its interest, but it also enhances the PU's performance.

\begin{figure}[t]
\begin{center}
\includegraphics[width=1\columnwidth]{Do.eps}
\caption{Optimal lower bound on $\mu_{\rm s}$ versus $\lambda_{\rm p}$ for different values of delay constraint.} \label{Fig6}
\end{center}
\vspace{-5mm}
\end{figure}

Furthermore, we investigate the effect of the QoS measures guaranteed for the PU on the throughput of the SU. Specifically, we depict the effect of varying the average primary queueing delay constraint on the SU throughput in Fig. \ref{Fig6}. The results are plotted for $q=0.8$ and $\lambda_{\rm e}=20$ energy packets/slot. From Fig. \ref{Fig5}, we note that the lower bound on SU throughput is enhanced as the primary delay constraint is relaxed, i.e., $D_{\rm \circ}$ is increased. This intuitive result is attributed to the fact that for a less restrictive primary delay constraint, the feasible set of the optimization variables is widened. Thus, we are able to achieve better SU throughput at higher values of $D_{\rm \circ}$.

In Fig. \ref{Fig7}, we show the essence of performing power allocation at the SU. We plot the resulting optimal lower bound on $\mu_{\rm s}$ with and without optimizing over the powers $P_{\rm s}^{\left(1\right)}$, $P_{\rm s}^{\left(2\right)}$ and $P_{\rm s}^{\left(3\right)}$. In the latter case, we set $P_{\rm s}^{\left(1\right)}$, $P_{\rm s}^{\left(2\right)}$ and $P_{\rm s}^{\left(3\right)}$ to their maximum possible value, i.e., $P_{\rm \circ}$. We set $\lambda_{\rm e}=20$ energy packets/slot and $q=0.8$.
From the figure, we note that the SU throughput is enhanced when we optimize over the transmission powers and the resulting optimal values for $P_{\rm s}^{\left(1\right)}$, $P_{\rm s}^{\left(2\right)}$ and $P_{\rm s}^{\left(3\right)}$ is definitely less than $P_{\rm \circ}$. Thus, we achieve higher SU throughput at lower average power consumption. We note that at $\lambda_{\rm p}=0.3759$ packets/slot, the problem becomes infeasible and there are no optimization variables would satisfy the primary constraints. Hence, the SU remains silent and it gains no access to the channel.

\begin{figure}[t]
\begin{center}
\includegraphics[width=1\columnwidth]{power.eps}
\caption{Effect of power allocation on SU throughput.} \label{Fig7}
\end{center}
\vspace{-5mm}
\end{figure}

\section{Conclusion}\label{conclusion}
We propose a novel random spectrum sensing and access scheme for a cognitive radio SU equipped with energy harvesting capability. The cognitive user utilizes the primary feedback signals to decide on its spectrum sensing and access probabilities. Furthermore, we optimize over the SU's transmission power with the objective of maximizing its throughput while simultaneously keeping the primary queue stable, and its average delay below a certain threshold. We investigate the effect of erroneous primary feedback signals overheard by the SU. We show that reliable feedback is in the interest of both the PU and SU. Moreover, we show the essence of performing power allocation at the SU compared to schemes in which SUs transmit with their maximum permissible power.

\section*{Appendix}
We derive the probability of successful transmission in the presence of interference. Consider two nodes, $\rm A$ and $\rm B$, transmitting to a common destination, $\rm d$, with average powers $P_{\rm A}$ and $P_{\rm B}$, respectively. With respect to $\rm d$, node's $\rm A$ transmission is the intended signal, while node $\rm B$ is considered an interferer. In a given time slot, $A$ transmits one packet ($\beta$ bits). The transmission duration is either $T-\tau$ or $T$ (seconds) depending on whether the transmission is preceded by a sensing phase or not, respectively. Thus, the transmission rate is given by
\begin{equation}
r_{\rm i}=\frac{\beta}{T-{\rm i}\tau}, ~{\rm i} \in \{0,1\}.
\end{equation}
Consequently, we use the notion of channel outage to write the probability of success on the link $\rm A \rightarrow d$ as
\begin{equation}
\mathbf{P}_{\rm \!i}(P_{\rm \!A},P_{\rm \!B})\!=\!\text{Pr}\! \left\{W\log\! \left[1\!+\!\frac{P_{\rm A}|h_{\rm A,d}|^2}{\mathcal{N}_{\circ}W+P_{\rm B}
|h_{\rm B,d}|^2}\right]\!>\!r_{\rm i}\right\}
\end{equation}
where $W$ is the channel bandwidth and $\text{Pr}\{\mathcal{E}\}$ denotes the probability of event $\mathcal{E}$. After some algebraic manipulation,
\begin{equation}
\mathbf{P}_{\rm \!i}(P_{\rm \!A},P_{\rm \!B})\!\!=\!\! \text{Pr}\! \left\{\! |h_{\rm A,d}|^2\!>\!\frac{2^{\frac{r_{\rm i}}{W}}-1}{P_{\rm \!A}/(\mathcal{N}_{\circ}W)}\!+\!\frac{(2^{\frac{r_{\rm i}}{W}}-1)P_{\rm \!B}|h_{\rm B,d}|^2}{P_{\rm \!A}}\! \right\}\!.
\end{equation}
For the ease of exposition, let $a=\frac{2^{\frac{r_{\rm i}}{W}}-1}{P_{\rm \!A}/(\mathcal{N}_{\circ}W)}$, $b=\frac{(2^{\frac{r_{\rm i}}{W}}-1)P_{\rm \!B}}{P_{\rm \!A}}$ and $X=|h_{\rm B,d}|^2$. Thus, we have
\begin{equation}\label{prob_success}
\mathbf{P}_{\rm \!i}(P_{\rm \!A},P_{\rm \!B})=\text{Pr} \left\{ |h_{\rm A,d}|^2>a+bX \right\}.
\end{equation}
Using total probability theory, (\ref{prob_success}) can be written as
\begin{equation}\label{total_prob}
\mathbf{P}_{\rm \!i}(P_{\rm \!A},P_{\rm \!B})=\int_{\rm 0}^{\infty} \text{Pr} \left\{ |h_{\rm A,d}|^2>a+bx \right\}
f_{\rm X}(x)dx
\end{equation}
where $f_{\rm X}(.)$ denotes the probability density function (PDF) of $X$. Following the channel model described in Section \ref{system_model}, $|h_{\rm A,d}|^2 \sim \text{exp}(1/\sigma_{\rm A,d}^{2})$ and $X \sim \text{exp}(1/\sigma_{\rm B,d}^{2})$. Therefore,
\begin{equation}\label{term_1}
\text{Pr} \left\{ |h_{\rm A,d}|^2>a+bx \right\}=e^{-(a+bx)/\sigma_{\rm A,d}^{2}}
\end{equation}
\begin{equation}\label{term_2}
f_{\rm X}(x)=\frac{1}{\sigma_{\rm B,d}^{2}}e^{-x/\sigma_{\rm B,d}^{2}}.
\end{equation}
Substituting by (\ref{term_1}) and (\ref{term_2}) in (\ref{total_prob}) and solving the integral, the probability of success on the link $A \rightarrow d$ in the presence of an interferer ($B$) is given by
\begin{equation}
\mathbf{P}_{\rm \!i}(P_{\rm \!A},P_{\rm \!B})=\frac{\sigma_{\rm A,d}^{2}~e^{-a/\sigma_{\rm A,d}^{2}}}{\sigma_{\rm A,d}^{2}+b~\sigma_{\rm B,d}^{2}}.
\end{equation}

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