text:
\begin{document}
\author {Yasha Savelyev}
\title   {Gromov $K$-area and jumping curves in $ \mathbb{CP} ^{n}$}
\maketitle
\begin{abstract} We give here some extensions of Gromov's and Polterovich's
theorems on $\karea$ of $ \mathbb{CP} ^{n}$, particularly in the symplectic and
Hamiltonian context. Our main methods involve Gromov-Witten theory, and some
connections with Bott periodicity, and loop groups. The argument is closely
connected with study of jumping curves in $ \mathbb{CP} ^{n}$,
and as an upshot we also prove a possibly new, general  theorem on these jumping curves.

\end{abstract}
\section {Introduction}
In \cite{Gromov} Gromov proposed an interesting way to probe the macroscopic
geometry of Riemannian and symplectic manifolds by means of geometry of complex
vector bundles on the manifold. This is a geometric analogue
of the idea of K-theory which is partly why it was named
$\karea$. This
construction involves minimizing sup norm of the curvature  over all
 homologically essential vector bundles and all connections, and his main
 theorem in the Riemannian setting for spin, positively curved manifolds $X,g$ used somewhat mysteriously the index theorem for the
twisted Dirac operator.

To pass to the symplectic world Gromov  considered an additional variation
over all compatible metrics, i.e. all compatible complex structures with a
symplectic form $\omega$ on $X$. At the moment the resulting invariant is still
very poorly understood. Here we focus on $X = \mathbb{CP} ^{n}$ and relate this
notion the classical theory of jumping curves and and quantum classes originally
defined by the author in \cite{GS}, and intertwined with Bott periodicity and aspects of
differential geometry of loop groups, \cite{BP}. Interestingly, this allows us
to arive at a purely algebraic geometric theorem in the theory of jumping curves,
as well at its symplectic generalization.

Some of the symplectic methods of the present paper continue in spirit upon
Polterovich \cite{polterovich}, and Entov
\cite{Entov},
but our viewpoint of looking at all vector bundles
simultaneously is closer to Gromov's original viewpoint. Although practically
this does not turn out to be very important here.

\subsection{}
We begin by discussing $\karea$ in the symplectic context.
Fix $$p: E \to X$$  a rank $r$ complex vector bundle
with $c_1 (E)=0$ over a closed symplectic manifold $(X ^{2n}, \omega)$ and let
$p: P \to X$  denote its projectivization. Let $ \mathcal {A}$ be a Hamiltonian
connection on $P$, with curvature 2-form $R ^{ \mathcal {A}}$, which at $x \in
X$ takes values in the Lie algebra of $ \text {Ham}( p ^{-1} (x))$, i.e. the space of normalized smooth
functions on $p ^{-1} (x)$. Here normalized means $$\int _{p
^{-1} (x)} G \Vol _{\omega _{st}} =0,$$ and $\omega _{st}$ on $p ^{-1} (x) \simeq \mathbb{CP} ^{r-1}$ is always assumed
to be the standard form with $\omega _{st} ( [line])=1$.

 Let $j_X$ be an
$\omega$-compatible almost complex structure on $X$,
 and $g _{j_X}$ the associated metric. We define the norm of curvature by
\begin{equation} \label {eq.norm} ||R ^{ \mathcal {A}}|| _{g _{j_X}} = \sup_ {x
\in X, \xi, \eta \in T_x X}  |R ^{ \mathcal {A}} (\xi, \eta)| ^{+}_H,
\end{equation}
where $| \cdot| ^{+}_H$ is ``half'' the Hofer norm $|G| ^{+}_H= \max G$, for $G:
(p^{-1} (x)
\simeq
\mathbb{CP} ^{r-1}) \to \mathbb{R}$ in the Lie algebra of $\text{Ham}(p ^{-1}
(x))$, and the supremum is over all orthonormal pairs $\xi, \eta$.

Here is the basic  quantity we will study:
\begin{equation} \label {eq.karea1}\karea ^{-1} (X, \omega) = \inf _{E,
\mathcal {A}, j_X} ||R ^{  \mathcal {A}}|| _{g _{j_X}},
\end{equation}
where the infinum is over all $E$ with some non-vanishing Chern number,
and $2r  \geq \dim _{ \mathbb{R}}X$.
\begin{remark} The quantity \eqref{eq.karea1} is closely
related to the one studied by Gromov in \cite{Gromov}. This relationship is
discussed in \cite{polterovich}. We make a few comments: Gromov does not
projectivize and works with unitary connections and consequently with standard norm on the Lie algebra of $U (r)$,
he also works with the inverse of our quantity, which we symbolize
 by the superscript $-1$ in $\karea ^{-1}$. The condition that
$2r  \geq \dim _{ \mathbb{R}}X$ is related to stability for homotopy
 groups of $SU (n)$ and is vacuous if we restrict to unitary connections, since
after stabilizing $E$ we may extend the unitary connection to the stabilization,
without affecting the norm \eqref{eq.norm}.
\end{remark}
Here is our first theorem:
\begin{theorem}  \label{thm.projective}
\begin{equation}  \karea ^{-1} (\mathbb{CP} ^{n}, E) \equiv \inf
_{\mathcal {A}, j _{X}} ||R ^{  \mathcal {A}}|| _{g _ {{j} _{X}}}  \geq 1,
\end{equation}
where the infinum is over all Hamiltonian connections $ \mathcal {A}$ on
projectivization of a fixed complex vector bundle $E$ on $ \mathbb{CP}
^{n}$, provided that $$\rank _{ \mathbb{C}} E \geq n,$$ $c_1(E)=0$ and some
other Chern class of $E$ does not vanish.
In particular $$\karea ^{-1} ( \mathbb{CP} ^{n}) \geq 1.$$
\end {theorem}
This answers a question of Polterovich in
\cite{polterovich} about finding bounds for $\karea ^{-1} ( \mathbb{CP} ^{2},
E)$, for rank $2$ complex vector bundle over $ \mathbb{CP} ^{2}$, with
$c_1(E)=0, c_2(E)=2$. If  $c_1$ does not vanish then the argument is more
elementary, and already worked out in \cite{polterovich}. Although one
can also adapt our discussion to subsume this case.

The above theorem extends:
  \begin{theorem} [Gromov,
\cite{Gromov}] \label{thm.gromov} \begin{equation*}  \karea_U ^{-1} (
\mathbb{CP}^{n}, \omega _{st}) \geq 1.
\end{equation*}
\end{theorem}
Here $U$ in $\karea_U ^{-1}$ emphasizes that Gromov worked with unitary
connections. Notice we have the same lower bound in the unitary and
Hamiltonian case.
\subsection {Jumping curves in $ \mathbb{CP} ^{n}$} Although the proof of
Theorem \ref{thm.projective} is via a rather transcendental piece of technology
known as quantum classes, it is also closely related  to the classical notion of
jumping curves in $ \mathbb{CP} ^{n}$. In fact, as a corollary, we obtain an
interesting phenomenon regarding these jumping curves. Here is a simplified
 version of the definition,  suitable in our context.
 \begin{definition} Let $E \to \mathbb{CP} ^{n}$ be a $rank _{ \mathbb{C}} E =r$
holomorphic vector bundle, with $c_1 (E)=0$. A smooth rational curve $C$ in $
\mathbb{CP} ^{n}$ will be called a \emph { \textbf{jumping curve}}, if the
restriction of $E$ to $C$ (by which we mean pullback) is not trivial as a
holomorphic vector bundle.
\end{definition}
Jumping curves $C$ can be further classified by the holomorphic isomorphism type
of $E| _{C}$, which by Grothendieck-Birkhoff theorem is:
\begin{equation} E| _{C} \simeq \bigoplus _{i} \mathcal {O} (\alpha (i)), \text
{ with } \sum _{i} \alpha (i)=0.
\end{equation}

We can actually give a symplectic generalization of this notion as follows:
Let $P _{E} \to \mathbb{CP} ^{n}$ denote the projectivization of $E$, which is a
Hamiltonian bundle and so its total space has a natural deformation class of
symplectic forms $\Omega$, extending the fiber wise symplectic forms $\omega
_{st}$, see \cite{MS2}.

For a rational curve $C$ in $ \mathbb{CP} ^{n}$, we have a smooth identification
of $P _{E}| _{C}$ with $\mathbb{CP} ^{r-1} \times S ^{2}$, well defined in
homology, since our structure group $ \text {Ham}( \mathbb{CP} ^{r-1}, \omega
_{st})$ acts trivially on homology.
 \begin{definition} Let $E \to \mathbb{CP} ^{n}$ be a complex vector bundle with
$c _{1} (E)=0$. Let $J$ be an almost complex structure compatible with $\Omega$
on $P _{E}$, and such that the projection map $P _{E} \to \mathbb{CP} ^{n}$ is
$J$-holomorphic. We will call such $J$ \emph { \textbf{admissible}}. A smooth
rational curve $C$ in $ \mathbb{CP} ^{n}$ is called a \emph { \textbf{jumping
curve}}  if $P _{E}| _{C} \simeq \mathbb{CP} ^{r-1} \times S ^{2}$ has a
$J$-holomorphic section in class $- d [line] + [S^2]$, $d > 0$.
\end{definition}
A more natural way of stating this, is that $P _{E}| _{C}$ has a $J$ holomorphic
section $u$, with $ \langle [\widetilde{\Omega}],  [u] \rangle = -d$, $d >0$,
where $[ \widetilde{\Omega}]$ is the coupling class of $P _{E}$, see \cite{MS2}.

When $J$ is induced by a holomorphic structure on $E$, this  notion is
equivalent to the classical notion, since in this case $P _{E}|_C$ is a
generalized Hirzebruch bundle:
\begin{equation*} P _{E}| _{C} = S ^{3} \times _{S ^{1}} \mathbb{CP} ^{r-1},
\end{equation*}
for some circle subgroup $S ^{1} \in SU (r)$. If this subgroup
is non-trivial and $H$ denotes it's generating Hamiltonian, then points $x \in F
_{\max}$, the maximum set of $H$, are fixed points of the $S ^{1}$ action on $
\mathbb{CP} ^{r-1}$, and give holomorphic sections $S ^{3} \times _{S ^{1}}
\{x\}$ of $P _{E}| _{C}$ with above $d >0$.

  Under some conditions
the locus set of classical jumping curves may be a divisor, see \cite{bundles}, but in general it is extremely
difficult to determine this locus. We will prove the following: \begin{theorem}
\label{thm.jumpline} Let $E \to \mathbb{CP} ^{n}$ be a rank $r \geq n$,
complex vector bundle, with $c_1 (E)=0$ and some other Chern class non-zero.
Then for any admissible $J$  on $P _{E}$ it has a degree one  jumping
curves. If we also assume that $J$ is suitably generic then $d$ above can be
chosen to be 1.
\end{theorem}
For emphasis we give this corollary.
\begin{corollary} Let $E
\to \mathbb{CP} ^{n}$ be a rank $r \geq n$ holomorphic vector bundle with $c_1
(E)=0$, and some other Chern class non-zero, then it has  jumping
curves.
\end{corollary}
\begin{remark} The condition $r \geq n$ can likely be dropped but for this would
require computation of quantum classes in the ``unstable'' range, which at the
moment is very technically difficult. However,  there may of course be
a less transcendental algebraic geometry approach.
\end{remark}

\subsection*{Acknowledgements} I am deeply grateful to Leonid Polterovich for
explaining to me his ideas on $\karea$, and compelling me to think about this
subject. I also thank Jarek Kedra and
Dusa McDuff and Leonid for discussions and comments, as well as the anonymous
referee for some interesting questions, and numerous comments. This paper was
primarily written during the author's stay at MSRI during spring of 2010, much thanks to the organizers and
administration for creating a pleasant atmosphere.
 \section {Setup} \label{section.setup} Our main tool are certain characteristic
 cohomology classes
 \begin{equation} \label {eq.qclasses.unstable}
 qc _{k} \in H ^{2k} (\Omega SU (r),
QH ( \mathbb{CP} ^{r-1})).
\end{equation}

These classes were originally defined and named
\emph{quantum classes} in much more generality in \cite{GS}.  Note however, that
$qc_k$ here is $qc _{2k}$ in \cite{GS}. The grading change is for convenience,
as in this context all odd quantum classes vanish.

We will not need the full definition, just some basic geometric content: and the
following theorem \cite{BP}:

   \begin{theorem} \label{theorem.corollary}
The classes $qc  _{k}$ on $\Omega SU (r)$ are algebraically independent and
generate cohomology in the stable range $2k \leq 2r-2$, with coefficients in
$QH ( \mathbb{CP} ^{r-1})$.
\end{theorem}

Here is a brief overview of the geometric construction of quantum
classes. For more details of the following discussion see \cite{GS}. Let
$M \hookrightarrow P \to X$ be a Hamiltonian bundle, with monotone fiber $(M,
\omega)$ over a closed, oriented, smooth manifold $X$, with a Hamiltonian
connection $ \mathcal {A}$. (Here $M$ is used to denote a symplectic manifold because it serves a different logical purpose to $(X, \omega)$ of Introduction,
but this $M$ will just be $ \mathbb{CP} ^{r-1}$ in the rest of the paper.) We
have a natural $S ^{1}$ action on $\Omega ^{2} X$, induced by rotation of  $S
^{2}$, along an axis of revolution containing the base point $0 \in S ^{2}$.

Let
$\Omega ^{2} X _{S ^{1}}$ denote the Borel $S ^{1}$ quotient:
\begin{equation} \Omega ^{2} X _{S ^{1}} = \Omega ^{2} X \times _{S ^{1}} S
^{\infty}. \end{equation}

Given
a cycle $$B \xrightarrow{f} \Omega ^{2} X _{S ^{1}} \xrightarrow{} \Omega ^{2} B \text {Ham}(M, \omega) _{S ^{1}},$$
 where the second map
 is induced by the classifying map $X \to B \text {Ham}(M,
\omega)$ for the above bundle, there is a naturally
induced Hamiltonian bundle $M \hookrightarrow P _{f} \to Y$, where $Y \to B$ is
an oriented $S ^{2}$-bundle over $B$. A quick explanation for this: any  map $B \to  \Omega ^{2} B
\text {Ham}(M, \omega) _{S ^{1}}$ is induced by a  map $Y \to B \text {Ham}(M,
\omega)$, where $Y$ is the $S ^2$-bundle over $B$, classified by the composition
$B \to \Omega ^{2} B \text {Ham}(M, \omega) _{S ^{1}} \to \mathbb{CP} ^{\infty}$, with
the map to $ \mathbb{CP} ^{\infty}$ being the canonical projection.
Equivalently we have a bundle $$M \times S ^{2} \to P _{f} \xrightarrow{p} B,$$ with $p$ denoting natural projection.

We may  define  classes $$qc_* \in H ^{*} (B, QH(M))$$ for $P _{f}$ as in
\cite{GS}, via count of certain $p$-fiberwise holomorphic curves, with a
$p$-fiberwise family of complex structures on $P _{f}$, induced by some
Hamiltonian connection $ \mathcal {A}$ on $M \hookrightarrow P \to X$. These
classes are induced by universal classes
\begin{equation}  \label {eq.qclasses} qc_* \in H ^{*} (\Omega ^{2} B\text
{Ham}(M, \omega)_ {S ^{1}}, QH(M)).
\end{equation}

\section {Proofs}

\begin{proof}[Proof of Theorem \ref{thm.jumpline}] Let $E$ be a rank $r$
complex vector bundle over $ \mathbb{CP} ^{n}$, with $n \leq r$, and some Chern
class non-zero. We may assume without loss of generality that $E$ has a
non-vanishing Chern number. (Otherwise,  restrict the following discussion to a
subspace $ \mathbb{CP} ^{i} \subset \mathbb{CP} ^{n}$, corresponding to a
non-zero class $c_i (E)$.) And let $$ \mathcal {M} _{0,1 }(\mathbb{CP} ^{n},
[line],j; x_0, [\mathbb{CP} ^{n-1}]) \to \mathbb{CP} ^{n},$$ denote the moduli
 space of curves with 1 free marked point and 2 fixed marked points mapping to
  $x_0$, $ \mathbb{CP} ^{n-1}$, $x_0 \notin \mathbb{CP} ^{n-1}$, with
   $$ev: \mathcal {M} _{0,1 }(\mathbb{CP} ^{n}, [line],j; x_0, [\mathbb{CP} ^{n-1}]) \to \mathbb{CP} ^{n}, $$
denoting the evaluation map given by evaluating at the free marked point. It is
well known that the standard complex structure on $ \mathbb{CP} ^{n}$ is regular
and that for this standard $j$ $ev$ is a degree one map \begin{equation*} P \to
\mathbb{CP} ^{n},
\end{equation*}
where $P$ is an $S ^{2}$-bundle over $ \mathbb{CP} ^{n-1}$ associated to the
Hopf bundle. This is because there is a unique complex line through a pair of
points in $ \mathbb{CP} ^{n}$. The induced cycle $f: \mathbb{CP} ^{n-1} \to
\Omega ^{2} \mathbb{CP} ^{n} _{ S ^{1}}$ represents a class denoted $a$.
Let $e: \mathbb{CP}
^{\infty} \to \Omega ^{2} BSU (r)_{S ^{1}} = \Omega ^{2} BSU (r) \times _{S ^{1}}
S ^{\infty}$ be the  section corresponding to the canonical fixed point of the $S
^{1}$ action on $\Omega ^{2} BSU (r)$, i.e. the constant map of $S ^{2}$ to the
based point $x_0 \in X$.
 \begin{lemma} \label{lemma.k-essential}   $$0 \neq f _{  {E}_*} a\in H _{2n-2} (\Omega ^{2}BSU (r) _{S ^{1}})/e_* H_* ( \mathbb{CP}
^{\infty}),$$ where $$f _{E}: \Omega ^{2} \mathbb{CP} ^{n}_{S ^{1}} \to \Omega
^{2}BSU (r)_{S ^{1}}$$ is the map induced by $ E \to \mathbb{CP} ^{n}$.
\end{lemma}

\begin {proof}
Let us suppose otherwise.

The composition map $$ev: P \to \mathbb{CP} ^{n} \to
BSU (r),$$ is non vanishing in homology, since $E$ has a non-vanishing Chern
number and $$ev: P  \to \mathbb{CP} ^{n}$$ is degree one by discussion above.

 Let $$H: T \to \Omega ^{2} BSU (r)_{S ^{1}}$$ be a bordism of
$f _{ E _{*}}a _{i-1}$ to $c \in e_*H_* ( \mathbb{CP} ^{\infty})$. We'll call
the corresponding boundary pieces of $T$ by $T _{a}$ and $T _{c}$.  Consequently,
the bordism $H$ induces an $S ^{2}$ bundle
$P _{T}$ over $T$,  it is the pull-back of the tautological $S
 ^{2}$ bundle $$(\Omega ^{2} _{x_0}BSU (r) \times E ^{\infty}) \times _{S ^{1}} S
 ^{2} \to\Omega ^{2} _{x_0} BSU (r)_{S ^{1}}.$$ and of course  $P _{T}$
restricts over $T _{a}$ to $P$. We have a natural ``evaluation'' map
\begin{equation} ev _{T}: P _{T} \to BSU (r),
\end{equation}
restricting to evaluation maps $ev$, $ev_c$ over boundary, and so a homology
of $[ev]$ to $[ev_c]$, but $ev_c$ is the constant map to the based point $x_0
\in BU$ a contradiction.
\end {proof}
\begin{lemma} \label{lemma.nonvanish}  For some $\alpha_i, \beta _{i}$
 \begin{equation} \label {eq.nonvanish2} \langle \prod _{i,j} qc ^{\alpha_i}
 _{\beta_i} \wedge l ^{j}, f _{{ E} _*} a _{n-1} \rangle \neq 0.
\end{equation}
\end{lemma}
\begin{proof} Note that all of cohomology of $\Omega ^{2} BSU (r) \simeq \Omega
SU (r)$, is in even degree, since by Milnor-Morre, Cartan-Serre \cite{MM},
\cite{CS}, the rational homology algebra is generated as a ring with Pontryagin
product by the  rational homotopy groups, (via Hurewicz homomorphism) which are
all in even degrees since the rational homotopy groups of $SU (r)$ are well known to be all in odd degrees. (In fact $SU (r)$ has the
rational homotopy type of the product of odd spheres $S ^{3} \times S ^{5}
\ldots$) Consequently,  the Serre spectral sequence degenerates at the second
page and so:
 \begin{equation} H ^{*} (\Omega ^{2} BSU (r) _{S ^{1}}) \simeq H ^{*} (\Omega ^{2}
BSU (r)) \otimes H ^{*} ( \mathbb{CP} ^{\infty}) \simeq H ^{*} (\Omega SU (r))
\otimes H ^{*} ( \mathbb{CP} ^{\infty}).
\end{equation}
 Our lemma then follows by  Lemma \ref{lemma.k-essential} and Theorem
 \ref{theorem.corollary}.
 \end{proof}
 The theorem then readily follows. Since by  construction of quantum
 classes and
 Lemma \ref{lemma.nonvanish},  for any fixed (not necessarily regular) complex
 structure $J$ on $P$ compatible with $\Omega$, and with projection to $
 \mathbb{CP} ^{n}$ for some complex line $l$ in $
 \mathbb{CP} ^{n}$, the restriction of $P$ to $l$, which is diffeomorphic to $ \mathbb{CP} ^{r-1}
 \times S ^{2}$ has a $J $ holomorphic stable section $u$ in total class $S= -
 [line] + S ^{2}$. As otherwise the relavant Gromov-Witten invariants in class
 $S$ all vanish and
 \eqref{eq.nonvanish2} is impossible. Of course the stable
 section $u$ may be in the form of a holomorphic section $u _{p}$ in class $- d[line] + S ^{2}$, with $d>1$ together with some vertical
 holomorphic bubbles, but this still implies our claim.
\end {proof}
\begin {proof} [Proof of Theorem \ref{thm.projective}]
We just need the following lemma:
\begin{lemma}  \label{lemma.norm} The norm of the curvature $||R _{ \mathcal
{A}}||$ of the projectivization $ \mathbb{CP} ^{r-1} \hookrightarrow P \to
\mathbb{CP} ^{n}$ is at least $1$.
\end{lemma}
\begin{proof}
 Let $ \widetilde{\Omega}$ denote the coupling form of the Hamiltonian fibration
 $P$ associated to $ \mathcal {A}$, (see for example \cite{MS2} for discussion on coupling forms). This  is a
certain closed form associated to  the curvature form of $ \mathcal {A}$,
with the following properties.

The
restriction of $ \widetilde{\Omega} $ to fibers $M \simeq \mathbb{CP} ^{r-1}$ of
$P \to \mathbb{CP} ^{n}$ coincides with $\omega _{st}$. The $
\widetilde{\Omega}$-orthogonal subspaces in $TP$ to the fibers are horizontal
subspaces, whose value on horizontal lifts $$ \widetilde{v}, \widetilde{w} \in T _{m,z} P$$ of $v,w \in T _{z} \mathbb{CP} ^{n}$
are given by $$ \widetilde{\Omega} ( \widetilde{v}, \widetilde{w}) = -R _{ \mathcal {A}} (v, w) (m),$$
for $m \in M _{z}$; in other words we evaluate the Lie algebra element of $ \text
{Ham}(M_z, \omega)$: $R _{ \mathcal {A}} (v, w)$ (i.e. a function on $M _{z}$)
at $m$.

Consider the symplectic form $\Omega =  \widetilde{\Omega}
+ (||R ^{ \mathcal {A}}|| + \epsilon) \omega _{st} ^{n}$, where $\omega _{st}
^{n}$, the standard Fubini-Study symplectic form on the base normalized by the
condition that the area of a complex line is 1, and $\epsilon>0$. Pick any
compatible complex structure $J _{ \mathcal {A}}$. By Theorem
\ref{thm.jumpline}, for some complex line $l$ in $ \mathbb{CP} ^{n}$,
the restriction of $P$ to $l$, which is diffeomorphic to $
\mathbb{CP} ^{r-1} \times S ^{2}$ has a $J _{ \mathcal {A}}$ holomorphic section
$u$ in class $S= -d \cdot [line] + S ^{2}$, with $d \geq 1$. Where $[line]$ is
the class of the complex line in $ \mathbb{CP} ^{r-1}$.

Since $J _{ \mathcal {A}}$ is $\Omega$ compatible, for the
class $S$, $J _{ \mathcal {A}}$-holomorphic section $u$ of $P|_l$ we get
\begin{equation} 0 \leq [\Omega] ( [u])  = [ \widetilde{\Omega }] ( [u])
+   ||R ^{ \mathcal {A}}|| + \epsilon.
\end{equation}
On the other hand $[\widetilde{\Omega}] = [\omega _{st}]$ on $P| _{l}$ since the
cohomology class of the coupling form is independent of the choice of
connection, and the form $\omega _{st}$ on $ \mathbb{CP} ^{r-1} \times S ^{2}$,
is another coupling form. Since $ [ \omega _{st}] ( [line])=1$ by our
normalization, it follows that $ [ \widetilde{\Omega} ] ([u])= -d$, since $[u] =
-[line]+ S ^{2}$. So we get: \begin{equation} d \leq  ||R ^{ \mathcal {A}}|| +
\epsilon,
\end{equation}
for every $\epsilon >0$.
\end{proof}
This finishes the proof of the theorem.
\end{proof}

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text:
\begin{document}
\title[Self-approximation of Dirichlet $L$-functions ]
{Self-approximation of Dirichlet $L$-functions}

\author{\sc Ram\={u}nas Garunk\v{s}tis}

\address{Ram\={u}nas Garunk\v{s}tis \\
Department of Mathematics and Informatics, Vilnius University \\
Naugarduko 24, 03225 Vilnius, Lithuania}

\thanks{Supported by grant No MIP-94 from the Research Council of Lithuania}

\email{ramunas.garunkstis@mif.vu.lt}

\urladdr{www.mif.vu.lt/~garunkstis}

\date{}

\setcounter{equation}{0}

\begin{abstract}
Let $d$ be a real number, let $s$ be in a fixed
compact set of the strip $1/2<\sigma<1$, and let
$L(s, \chi)$ be the Dirichlet $L$-function. The
hypothesis  is that for any real number $d$ there
exist `many' real numbers $\tau$ such that  the
shifts
 $L(s+i\tau, \chi)$ and $L(s+id\tau, \chi)$ are `near' each other. If $d$ is an algebraic irrational number then this was obtained by T. Nakamura.  \L. Pa\'nkowski solved the case then $d$ is a transcendental number. We prove the case then $d\ne0$ is a rational number.  If $d=0$ then by B. Bagchi we know that the above hypothesis is equivalent to the Riemann hypothesis for the given Dirichlet $L$-function. We also consider a more general  version of the above problem.
\end{abstract}

\maketitle

\section{Introduction}

Let, as usual, $s=\sigma+it$ denote a complex
variable. For $\sigma>1$, the Dirichlet
$L$-function is given by
\[
L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s},
\]
where $\chi(n)$ is a Dirichlet character $\bmod\,
q$.  For $q=1$ we get $L(s,\chi)=\zeta(s)$, where
$\zeta(s)$ is the Riemann zeta-function.

In \cite{bohr} Bohr proved that if $\chi$ is a
nonprincipal character, then the Riemann
hypothesis for $L(s, \chi)$ is equivalent to the
almost periodicity of $L(s, \chi)$ in the half
plane $\sigma>1/2$. A function $f(s)$ is  {\it
almost periodic} in a region $E\subset\Bbb C$ if
for any positive $\eps$ and any compact subset
$K$ in $E$ there exists a sequence of real
numbers $\dots<\tau_{-1}<0<\tau_1<\tau_2<\dots$
such that
$$\liminf_{m\to\pm\infty}(\tau_{m+1}-\tau_m)>0,\qquad\limsup_{m\to\pm\infty}\frac{\tau_m}m<\infty$$
and
$$|f(s+i\tau_m)-f(s)|<\eps\quad\textup{for all}\ s\in K \ \textup{and}\ m\in\Bbb Z$$
 hold.
Bohr \cite{bohr} also obtained that every
Dirichlet series is almost-periodic in its
half-plane of absolute convergence. Effective
upper bounds for the almost periodicity of
Dirichlet series with Euler products in the
half-plane of absolute convergence were
considered by Girondo and Steuding \cite{gs}.
Note that  every Dirichlet $L$-function is almost
periodic in the sense of Besicovitch on any
vertical line of the strip $1/2<\sigma<1$. For
this and related results see Besicovitch
\cite{bes} and Mauclaire \cite{mauc},
\cite{mauc2}.

Bagchi \cite{bag} proved that the Riemann
hypothesis for $L(s, \chi)$ ($\chi$  is an
arbitrary Dirichlet character) is true if and
only if for any compact subset $\mathcal K$ of
the strip $1/2<\sigma<1$ and for any $\eps>0$
\begin{align}\label{strong}
\liminf_{T\to\infty}\frac1
T\text{meas}\left\{\tau\in[0, T] :
\max_{s\in\mathcal K}|L(s+i\tau, \chi)-L(s,
\chi)|<\eps\right\}>0,
\end{align}
where $\text{meas}\, A$ stands for the Lebesgue
measure of a measurable set $A$. Bagchi says that
the Dirichlet $L$-function $L(s, \chi)$ is {\it
strongly recurrent} on the strip
$\sigma_0<\sigma<\sigma_1$ if  (\ref{strong}) is
valid for any compact $\mathcal K$ of the strip
$\sigma_0<\sigma<\sigma_1$. The strong recurrence
is connected with the universality property of
Dirichlet series. More about the universality and
the strong recurrence see Bagchi  \cite{bagdis},
\cite{bag}, \cite{bag87}, and Steuding
\cite{steu}.

There are several unconditional results
concerning the self-approximation of Dirichlet
$L$-functions in the critical strip. Let
$\mathcal K$ be a compact subset of the strip
$1/2<\sigma<1$ and let $\lambda\in\Bbb R$ be such
that $\mathcal K$ and  $\mathcal K+i\lambda
:=\{s+i\lambda : s\in\mathcal K\}$ are disjoint.
From Kaczorowski, Laurin\v cikas and Steuding
\cite{kls} it follows that for any character
$\chi$ and any $\eps>0$
\begin{align*}
\liminf_{T\to\infty}\frac1
T\text{meas}\left\{\tau\in[0, T] :
\max_{s\in\mathcal K}|L(s+i\lambda+i\tau,
\chi)-L(s+i\tau, \chi)|<\eps\right\}>0.
\end{align*}
Nakamura \cite{nak} considered the joint
universality of shifted Dirichlet $L$-functions.
His Theorem 1.1 leads to the following statement.
If $1=d_1, d_2, \dots, d_m$ are algebraic real
numbers {\it linearly independent} over $\Bbb Q$,
then for any Dirichlet character $\chi$ and any
$\eps>0$
\begin{align}\label{naka}
\liminf_{T\to\infty}&\frac1
T\text{meas}\left\{\tau\in[0, T] \right.
:\\&\left. \max_{1\le j,k\le m}\max_{s\in\mathcal
K}|L(s+id_j\tau, \chi)-L(s+id_k\tau,
\chi)|<\eps\right\}>0.\nonumber
\end{align}
If $m=2$ then Pa\'nkowski \cite{pan} using Six
Exponentials Theorem showed that (\ref{naka})
holds for $d_1, d_2$ are real numbers linearly
independent over $\Bbb Q$.

We prove the following theorem.
\begin{Theorem}\label{th}
Let $1=d_1, d_2, \dots, d_m$ be nonzero algebraic
real numbers and let $\mathcal K$ be a compact
subset of the strip $1/2<\sigma<1$. Then for any
Dirichlet character $\chi$ and any $\eps>0$ the
inequality (\ref{naka}) is valid.
\end{Theorem}
Note that Theorem \ref{th} remains true if $d_1,
d_2, \dots, d_m$ are replaced by $dd_1, dd_2,
\dots, dd_m$, where $d\in\Bbb R$. The next
theorem shows that `$\liminf $' in the inequality
(\ref{naka})  often can be replaced by `$\lim$'.
\begin{Theorem}\label{th2}
Let $d_1, d_2, \dots, d_m$ be any real numbers,
let $\chi_1, \chi_2, \dots,\chi_m$ be  any
Dirichlet characters, and  let $\mathcal K$ be a
compact subset of the strip $1/2<\sigma<1$. Then
for any $\eps>0$, except an at most countable set
of $\eps$, there exists a limit
\begin{align*}
\lim_{T\to\infty}&\frac1
T\text{meas}\left\{\tau\in[0, T] : \max_{1\le
j,k\le m}\max_{s\in\mathcal K}|L(s+id_j\tau,
\chi_j)-L(s+id_k\tau,
\chi_k)|<\eps\right\}.\nonumber
\end{align*}
\end{Theorem}

The mentioned results of Nakamura and Pa\'nkowski
together with Theorem \ref{th} and Theorem
\ref{th2} lead to the following corollary.

\begin{Corollary}\label{cor}
Let $d$ be a nonzero real number and let
$\mathcal K$ be a compact subset of the strip
$1/2<\sigma<1$. Then for any Dirichlet character
$\chi$ and any $\eps>0$, except an at most
countable set of $\eps$,
\begin{align}\label{lim}
\lim_{T\to\infty}\frac1
T\text{meas}\left\{\tau\in[0, T] :
\max_{s\in\mathcal K}|L(s+i\tau,
\chi)-L(s+id\tau, \chi)|<\eps\right\}>0.
\end{align}
\end{Corollary}

From the proof of Theorem \ref{th2} we see that
for any real numbers $d_1,\dots,d_m$ and for any
Dirichlet characters $\chi_1,\dots,\chi_m$ the
function
\[
 g(\tau)=
\max_{1\le j,k\le m}\max_{s\in\mathcal
K}\left|L(s+id_j\tau, \chi_j)-L(s+id_k\tau,
\chi_k) \right|
\]
is Besicovitch almost periodic function (for the
definition see Section \ref{secproof} above the
proof of Theorem \ref{th2}). Let $\eps>0$ be such
that the limit (\ref{lim}) exists. For such $\eps$ we
define a characteristic function
\(I_\eps(\tau)\), $\tau\in\Bbb R$, by
\begin{align}
I_\eps(\tau)=\left\{
\begin{matrix}
1,&\text{if}&g(\tau)<\eps,\\
0,&\text{if}&g(\tau)\ge\eps.
\end{matrix}
\right.
\end{align}
It is known (Jessen and A. Wintner \cite[Section
12]{jw})
 that \(I_\eps(\tau)\) is
Besicovitch almost periodic function also. Thus
we can say that self-approximations of Dirichlet
$L$-functions, considered in this paper,
 usually appear in a regular way.

 Theorem \ref{th} and Theorem \ref{th2} are
proved in Section \ref{secproof}. Next we state
several lemmas.

\section{Lemmas}

We start from the following statement.
\begin{Lemma}\label{03}
Let $\mathcal K$ be a compact subset of the
rectangle $U$. Let
$$d=\min_{z\in\partial\, U}\min_{s\in \mathcal K}|s-z|.$$
If $f(s)$ is analytic on $U$ and
$$\int_{U}\left|f(s)\right|^2 d\sigma dt\le\eps,$$
then
$$\max_{s\in \mathcal K}\left|f(s)\right|\le\frac{\sqrt{\eps/\pi}}d.$$
\end{Lemma}
\proof The lemma can be found in Gonek \cite{gon}
(Lemma 2.5).
\endproof

\begin{Lemma}\label{13}
Let $a_1,\dots,a_N$ be real numbers linearly
independent over the rational numbers. Let
$\gamma$ be a region of the $N$-dimensional unit
cube with volume $V$ (in the Jordan sense). Let
$I_\gamma(T)$ be the sum of the intervals between
$t=0$ and $t=T$ for which the point
$(a_1t,\dots,a_Nt)$ is$\mod 1$ inside $\gamma$.
Then
$$\lim_{T\to\infty}\frac{I_\gamma(T)}{T}=V.$$
\end{Lemma}
\proof This is Theorem 1 in Apendix, Section 8,
of Voronin and Karatsuba \cite{vor}.
\endproof

For a curve $\omega(t)$ in $\Bbb R^N$ we
introduce the notation
$$\left\{\omega(t)\right\}=\left(\omega_1(t)-[\omega_1(t)],\dots,\omega_N(t)-[\omega_N(t)]\right),$$
where $[x]$ denotes the integral part of
$x\in\Bbb R$.
\begin{Lemma}\label{04}
Suppose that the curve $\omega(t)$ is uniformly
distributed$\mod 1$ in $\Bbb R^N$. Let $D$ be a
closed and Jordan measurable subregion of the
unit cube in $\Bbb R^N$and let $\Omega$ be a
family of complex-valued continuos functions
defined on $D$. If $\Omega$ is uniformly bounded
and equicontinuous, then
$$\lim_{T\to\infty}\frac{ 1}{T}\int_0^T f\left(\{\omega(t)\}\right){ 1}_D(t)dt=\int_D f(x_1,\dots,x_N) d x_1\dots d x_N$$
uniformly with respect to $f\in\Omega$, where
$1_D(t)$ is equal to $1$ if $\omega(t)\in D \mod
1$, and $0$ otherwise.
\end{Lemma}
\proof The lemma is Theorem 3 in  Appendix,
Section 8, of Voronin and Karatsuba \cite{vor}.
\endproof

\begin{Lemma}\label{indep}
Let $p_n$ be the $n$th prime number and $1=d_1,
d_2,\dots,d_l$ be algebraic real numbers which
are linearly independent over $\Bbb Q$. Then the
set $\{d_k\log p_n\}_{n\in\Bbb N}^{1\le k\le l}$
is linearly independent over $\Bbb Q$.
\end{Lemma}
\proof This is Proposition 2.2 in Nakamura
\cite{nak}. The proof is based on Baker's
\cite[Theorem 2.4]{bak} result.
\endproof

\section{Proof of  Theorem \ref{th} and Theorem \ref{th2}}\label{secproof}

\proof[Proof of Theorem \ref{th}] We define a
truncated Dirichlet $L$-function
$$L_v(s, \chi)=\prod_{p\le v}\left(1-\frac{\chi(p)}{p^s}\right)^{-1}.$$
Roughly speaking, we first prove Theorem \ref{th}
for the truncated Dirichlet $L$-function and
later we show that the tail is small.

 Let $\{d_1,
d_2,\dots, d_l\}$ be a maximal linearly
independent (over $\Bbb Q$) subset of the set
$\{d_1, d_2,\dots, d_m\}$. Then there are
integers $a\ne0$ and $a_{k,1}, a_{k,2},\dots,
a_{k,l}$  such that
\begin{align}\label{lin}
 d_k=\frac1 a\left(a_{k,1}d_1+a_{k,2}d_2+\dots+a_{k,l}d_l\right) \quad\textup{for}\quad l< k\le m.
\end{align}

Let
$$A=\max_{{l<k\le m }}\{|a_{k, 1}|+|a_{k, 2}|+\dots+|a_{k, l}|\}.$$
 Denote by
$\Vert x\Vert$ the minimal distance of $x\in\Bbb
R$ to an integer. If
\begin{align}\label{tau}
\left\Vert\tau\frac{d_n\log p}{2\pi
a}\right\Vert<\delta \quad\textup{for}\quad p\le
v\ \textup {and}\ 1\le n\le l
\end{align}
then
\begin{align*}
\left\Vert\tau\frac{d_n\log p}{2\pi
}\right\Vert<a\delta \quad\textup{for}\quad p\le
v\ \textup {and}\ 1\le n\le l
\end{align*}
and, by the relation (\ref{lin}),
$$\left\Vert \tau\frac{d_k\log p}{2\pi}\right\Vert<A\delta \quad\textup{for}\quad p\le v\ \textup {and}\ l< k\le m.$$
 By this and by
the continuity of the function $L_v(s, \chi)$ we
have that for any $\eps>0$ there is $\delta>0$
such that for $\tau$ satisfying (\ref{tau})
 \begin{align}\label{maxmax}
 \max_{1\le k,n\le m}\max_{s\in\mathcal K}
 \left|\log L_v(s+id_k\tau, \chi)
 -\log L_v(s+id_n\tau, \chi)\right|<\eps.
 \end{align}

For  positive numbers $\delta$, $v$, and $T$ we
define the set
\begin{align}\label{st}
S_T=S_T(\delta, v)=\left\{\tau : \tau\in[0,T],
\left\|\tau\frac{d_n\log p}{2\pi
a}\right\|<\delta, \
 p\le v, \ 1\le n \le l\right\}.
\end{align}
Let $U$ be an open bounded rectangle with
vertices on the lines $\sigma=\sigma_1$ and
$\sigma=\sigma_2$, where
$1/2<\sigma_1<\sigma_2<1$, such that the set
$\mathcal K$ is in $U$. Let $y>v$. We have
\begin{align*}
&\frac1T\intl_{S_T}\intl_{U}\sum_{k=1}^m\left|
\log L_y(s+id_k\tau, \chi)- \log L_v(s+id_k\tau,
\chi)\right|^2 d\sigma dt d\tau
\\ &=\sum_{k=1}^m\intl_{U}\frac1T\intl_{S_T}\left|\log L_y(s+id_k\tau, \chi)- \log L_v(s+id_k\tau, \chi)\right|^2 d\tau d\sigma
dt.
\end{align*}
For the inner integrals of the right-hand side of
the last equality  we will apply Lemma \ref{04}.
Let $p_n$ be the $n$th prime number. There are
indexes $M$ and $N$ such that $p_M\le v<p_{M+1}$
and   $p_N\le y<p_{N+1}$. By  generalized
Kronecker's theorem (Lemma \ref{13}) and by Lemma
\ref{indep} the curve
$$\omega(\tau)=\left(\tau \frac{d_k\log p_n}{2\pi a}\right)_{1\le n\le N}^{1\le k\le l}$$
is uniformly distributed$\mod1$ in $\Bbb R^{lN}$.
Let $R'$ be a subregion of the $lN$-dimensional
unit cube defined by inequalities
$$\|y_{k,n}\|\le\delta\quad\text{for}\quad 1\le k\le l\ \text{and}\ 1\le n\le M$$
 and
$$\left|y_{k,n}-\frac12\right|\le\frac12 \quad\text{for}\quad 1\le k\le l\ \text{and}\ M+1\le n\le N.$$
 Let $R$ be a subregion of the $lM$-dimensional unit cube defined by inequalities
$$\|y_{k,n}\|\le\delta\quad\text{for}\quad 1\le k\le l\ \text{and}\ 1\le n\le M.$$
Clearly
$$\operatorname{meas}R'=\operatorname{meas}R=(2\delta)^{lM}.$$
Note that
\begin{align}\label{logL}
&\log L_y(s+id_k\tau, \chi)- \log L_v(s+id_k\tau,
\chi)=\log\frac{L_y}{L_v}(s+id_k\tau,
\chi)\nonumber
\\
&=-\sum_{v<p\le
y}\log\left(1-\frac{\chi(p)}{p^{s+id_k\tau}}\right)=\sum_{v<p\le
y}\sum_{j=1}^\infty\frac{\chi^j(p)}{jp^{j(s+id_k\tau)}}
\\
&=\sum_{M<n\le
N}\sum_{j=1}^\infty\frac{\chi^j(p_n)}{jp_n^{j(s+id_k\tau)}}.\nonumber
\end{align}
Thus in view of the linear dependence (\ref{lin})
we get
\begin{align*}
&\lim_{T\to\infty}
\frac1T\int_{S_T}\sum_{k=1}^m\left|\log\frac{L_y}{L_v}(s+id_k\tau,
\chi)\right|^2 d\tau
\\
& =
\lim_{T\to\infty}\frac1T\int_{S_T}\left(\sum_{k=1}^l\left|\log\frac{L_y}{L_v}(s+id_k\tau,
\chi)\right|^2\right.
\\
&
\left.\phantom{=}+\sum_{k={l+1}}^m\left|\log\frac{L_y}{L_v}\left(s+\frac{i}{a}(a_{k,1}d_1+a_{k,2}d_2+\dots+a_{k,l}d_l)\tau,
\chi\right)\right|^2\right) d\tau.
\end{align*}
By Lemma \ref{04} and equality (\ref{logL}) we
obtain that the last limit is equal to
\begin{align*}
& \int_{R'}\left(\sum_{k=1}^l\left|\sum_{M<n\le
N}\sum_{j=1}^\infty\frac{\chi^j(p)e^{-2\pi
ijay_{k,n}}}{jp_n^{js}}\right|^2 \right.
\\&
\left.+\sum_{k=l+1}^m\left|\sum_{M<n\le
N}\sum_{j=1}^\infty\frac{\chi^j(p)e^{2\pi
ij(a_{k,1}y_{1,n}+a_{k,2}y_{2,n}+\dots+a_{k,l}y_{l,n})}}{jp_n^{js}}\right|^2\right)dy_{1,1}\dots
dy_{l,N}
\\
& =\operatorname{meas}
R\intl_0^1\dots\intl_0^1\left(\sum_{k=1}^l\left|\sum_{M<n\le
N}\sum_{j=1}^\infty\frac{\chi^j(p)e^{-2\pi
ijay_{k,n}}}{jp_n^{js}}\right|^2 \right.
\\&
\left.+\sum_{k=l+1}^m\left|\sum_{M<n\le
N}\sum_{j=1}^\infty\frac{\chi^j(p)e^{2\pi
ij(a_{k,1}y_{1,n}+a_{k,2}y_{2,n}+\dots+a_{k,l}y_{l,n})}}{jp_n^{js}}\right|^2\right)dy_{1,M+1}\dots
dy_{l,N}
\\
&= m\operatorname{meas} R \sum_{v<p\le
y}\sum_{j=1}^\infty\frac{1}{jp^{2j\sigma}}\ll
\operatorname{meas} R
\sum_{p>v}\frac{1}{p^{2\sigma}}.
\end{align*}
Consequently
\begin{align}\label{iint}
&\frac1T\intl_{S_T}\intl_{U}\sum_{k=1}^m\left|
\log L_y(s+id_k\tau, \chi)- \log L_v(s+id_k\tau,
\chi)\right|^2 d\sigma dt d\tau
\\
& \ll \operatorname{meas} R
\sum_{p>v}\frac{1}{p^{2\sigma_1}}.\nonumber
\end{align}
 Again, by generalized
Kronecker's theorem (Lemma \ref{13}),
\begin{align}\label{kron}
\lim_{T\to\infty}\frac1T\operatorname{meas}
S_T=\operatorname{meas} R.
\end{align}
By (\ref{iint}) and (\ref{kron}), for large $v$,
as $T\to\infty$, we have
\begin{align*}
\operatorname{meas}\left\{\tau : \tau\in S_T,
\int_{
U}\sum_{k=1}^m\left|\log\frac{L_y}{L_v}(s+id_k\tau,
\chi)\right|^2 d\sigma
dt<\sqrt{\sum_{p>v}\frac{1}{p^{2\sigma_1}}}\right\}
>\frac12T\operatorname{meas}R.
\end{align*}
Then Lemma \ref{03} gives
\begin{align*}
&\operatorname{meas}\left\{\tau : \tau\in S_T,
\max_{s\in\mathcal
K}\sum_{k=1}^m\left|\log\frac{L_y}{L_v}(s+id_k\tau,
\chi)\right|^2
d\tau\le\frac1{d\sqrt{\pi}}\left(\sum_{p>v}\frac{1}{p^{2\sigma_1}}\right)^\frac14\right\}
\\
&>\frac12T\operatorname{meas}R,
\end{align*}
where $d=\min_{z\in\partial\, U}\min_{s\in
\mathcal K}|s-z|.$ By the continuity of the
logarithm we obtain that for  any $\eps>0$ there
is $v=v(\eps)$ such that for any $y>v$
\begin{align}\label{maxsum}
&\operatorname{meas}\left\{\tau : \tau\in S_T,
\max_{s\in\mathcal
K}\sum_{k=1}^m\left|L_y(s+id_k\tau,
\chi)-L_v(s+id_k\tau, \chi)\right|^2
d\tau<\eps\right\}
\\
&>\frac12T\operatorname{meas}R.\nonumber
\end{align}
Now we will prove that for any $\delta>0$ there
is $y=y(\delta)$ such that
\begin{align}\label{maxsum2}
&\operatorname{meas}\left\{\tau : \tau\in [0,T],
\max_{s\in\mathcal
K}\sum_{k=1}^m\left|L(s+id_k\tau,
\chi)-L_y(s+id_k\tau, \chi)\right|^2
d\tau<\delta\right\}
\\
&>(1-\delta)T.\nonumber
\end{align}
The last formula together with (\ref{maxmax}),
(\ref{st}) and (\ref{maxsum}) yields
Theorem~\ref{th}. We return to the proof of
(\ref{maxsum2}). By the mean value theorem of the
Dirichlet $L$-function (Steuding \cite{steu},
Corollary 6.11) and by Carlson's Theorem
(Titchmarsh \cite{titch}, Chapter 9.51) we obtain
\begin{align*}
\lim_{T\to\infty}\frac1T\intl_{0}^{T}\left|L(s+ix\tau,
\chi)-L_{y}(s+ix\tau, \chi)\right|^2
 d\tau=\sum_{n>y}\frac{|\chi(n)|}{n^{2\sigma}},
\end{align*}
where $x$ is fixed. Thus (\ref{maxsum2}) follows
in view of
\begin{align*}
\intl_{0}^{T}\intl_{U}\sum_{k=1}^m\left|L(s+id_k\tau,
\chi)-L_{y}(s+id_k\tau, \chi)\right|^2 d\sigma dt
d\tau\ll
\sum_{n>y}\frac{|\chi(n)|}{n^{2\sigma_1}}.
\end{align*}
Theorem \ref{th} is proved.
\endproof
The proof of Theorem \ref{th2} is based on the
ideas of Mauclaire \cite{mauc}, \cite{mauc2}. It
uses the theory of Besicovitch almost periodic
functions. We recall related definitions.

Let
$$P(\tau)=\sum_{n\in F} a_ne^{i\lambda_n \tau},$$
where $F$ is a finite set, $\lambda_n$ are any
real numbers, and the coefficients $a_n$ are any
complex numbers. For real  $\tau$ we say that
$P(\tau)$ is a {\it trigonometric polynomial}.

A  function $f : \Bbb R\to\Bbb C$ is called {\it
uniformly almost periodic} ($U.A.P.$) if given
any $\eps>0$, there exists a trigonometric
polynomial $P(\tau)$ such that
\[\sup_{\tau\in\Bbb R}|f(\tau)-P(\tau)|\le\eps.\]

A  function $f : \Bbb R\to\Bbb C$ is called {\it
$B^q$  almost periodic} ($B^q.A.P.$), $q\ge1$, if
given any $\eps>0$, there exists a trigonometric
polynomial $P(\tau)$ such that
\begin{align}\label{bap}
\limsup_{T\to\infty}\frac1{2T}
\int_{-T}^{T}|f(\tau)-P(\tau)|^q d\tau\le\eps.
\end{align}
If $q=1$ then we write $B.A.P.$ (Besikovitch
almost periodic) instead of $B^1.A.P.$ For any
$q\ge1$ it is clear that every $U.A.P.$ function
is $B^q.A.P.$ and that every $B^q.A.P.$ function
is $B.A.P.$

\proof[Proof of Theorem \ref{th2}] Let
\[
 g(\tau)=
\max_{1\le j,k\le m}\max_{s\in\mathcal
K}\left|L(s+id_j\tau, \chi_j)-L(s+id_k\tau,
\chi_k) \right|
\]
  and let
$$F_{T}(x)=\frac{1}{T}\textup{meas}\left\{\tau\in[0, T] : g(\tau)<x\right\}$$
be a distribution function of $ g(\tau)$. If
$g(\tau)$ is $B.A.P.$ then it is known (see
Jessen and Wintner \cite[Theorem 27]{jw} or
Laurin\v cikas \cite[Theorem 6.3, Chapter
2]{laur}) that there is a distribution function
$F(x)$ such that $F_{T}(x)$ converges weakly to
$F(x)$ for $T\to \infty$. It means that if $F(x)$
is continuous at $x=\eps$ then
$$\lim_{T\to\infty} F_T(\eps)$$
exists. Thus to obtain Theorem \ref{th2} we need
to show that $g(\tau)$ is $B.A.P.$

We remark that if $a(t)$ and $b(t)$ are both
non-negative $B.A.P.$ functions of $t$, then,
$t\mapsto \max (a(t),b(t))$ is also $B.A.P.$
since $\max (a(t),b(t))$ can be written as
\begin{align*}
\max (a(t),b(t))=\frac{1}{2}\left( \left|
a(t)-b(t)\right| +(a(t)+b(t))\right) ,
\end{align*}
and the modulus of $B.A.P.$ function is again
$B.A.P.$ By this we have that  $g(\tau)$ is
$B.A.P.$ if the function
$$f(\tau)=\max_{s\in\mathcal
K}\left|L(s+id_1\tau, \chi_1)-L(s+id_2\tau,
\chi_2)\right|$$ is $B.A.P.$ In view of the note
below the formula (\ref{bap}) the function
$f(\tau)$ is $B.A.P.$ if there are
 $U.A.P$ functions $f_N(\tau)$ such that
\begin{align}\label{uap}
 \lim_{N \to +\infty }\left( \lim
\sup_{T\rightarrow +\infty
}\frac{1}{2T}\int_{-T}^{T}\left|
f(\tau)-f_N(\tau)\right|^{2}d\tau\right) =0.
\end{align}

Let
$$L_N(s,\chi)=\sum _{n\le N}\frac{\chi(n)}
{n^s}$$ be a partial sum of the Dirichlet series
associated with $L(s,\chi)$. Next we show that
the equality (\ref{uap}) is true with
\[f_N(\tau)=\max_{s\in\mathcal
K}\left|L_N(s+id_1\tau,\chi_1)- L_N(s+id_2\tau,
\chi_2)\right|.\]

 By repeating the proof of
Proposition 12 of Mauclaire \cite{mauc} we get
that $f_N(\tau)$ is $U.A.P.$ for any
$d_1,d_2\in\Bbb R$. Note that the case when $d_1$
or $d_2$ is equal to zero is already included in
Proposition 12 of Mauclaire \cite{mauc}.

Further we have that
\begin{align*}
&L(s+id_1\tau,\chi_1)-L(s+id_2\tau,\chi_2)\\
&=\left(L(s+id_1\tau,\chi_1)-L_N(s+id_1\tau,\chi_1)+L_N(s+id_2\tau,\chi_2)-L(s+id_2\tau,\chi_2)\right)\\
&\phantom{=}+\left(L_N(s+id_1\tau,\chi_1)-L_N(s+id_2\tau,\chi_2)\right),
\end{align*}
and as a consequence, we get that
\begin{align*}
&\left| f(\tau) -f_N(\tau) \right| \\
&\leq
\sup_{s\in K}\left|
L(s+id_1\tau,\chi_1)-L_N(s+id_1\tau,\chi_1)+L_N(s+id_2\tau,\chi_2)-L(s+id_2\tau,\chi_2)\right|
\\ &\le \sup_{s\in K}\left|
L(s+id_1\tau,\chi_1)-L_N(s+id_1\tau,\chi_1)\right|\\
&\phantom{\le}+\sup_{s\in
K}\left|L_N(s+id_2\tau,\chi_2)-L(s+id_2\tau,\chi_2)\right|.
\end{align*}
Then, in view of the inequality
$(a+b)^2\le2a^2+2b^2$, we obtain that
\begin{align*}
&\frac{1}{2T}\int_{-T}^{T}\left| f(\tau) -f_N(\tau) \right| ^{2}dt \\
&\leq \frac{1}{T}\int_{-T}^T\left( \sup_{s\in
K}\left| L(s+id_1\tau,\chi_1)-L_N(s+id_1\tau,\chi_1)\right| \right)
^{2}dt\\&\phantom{\le}+\frac{1}{T}\int_{-T}^T\left(
\sup_{s\in K}\left|
L(s+id_2\tau,\chi_2)-L_N(s+id_2\tau,\chi_2)\right| \right) ^{2}dt.
\end{align*}
By Mauclaire \cite[Theorem 5.1]{mauc2} we have
that, for any real $d$,
\[
\lim_{N \to +\infty }\left( \lim
\sup_{T\rightarrow +\infty }\frac{1}{2T^{\prime
}}\int_{-T}^{T}\left( \sup_{s\in K}\left|
f(s+idt)-f_N(s+idt)\right| \right) ^{2}dt\right)
=0.
\]
This proves the equality (\ref{uap}) and Theorem \ref{th2}
\endproof

From the proof we see that Theorem \ref{th2}
remains true with Dirichlet $L$-functions
$L(s,\chi_j)$, $j=1,\dots,m$,  replaced by any
general Dirichlet series satisfying conditions of
Theorem 5.1 of Mauclaire \cite{mauc2}.

{\bf Acknowledgment.} We thank Jean-Loup
Mauclaire for suggesting Theorem~\ref{th2} and
for other useful comments which helped to improve
the paper.

{\bf Remark.} The `$\liminf$' version of
Corollary \ref{cor} is  independently obtained by
Takashi Nakamura in ``The generalized strong
recurrence for non-zero rational parameters",
arXiv:1006.1778v1 [math.NT].


TITLE: Question about "Magnifying" an inequality for big-oh analysis
QUESTION [0 upvotes]: Here's an example directly from my numerical analysis text book:
$$
a_n = \frac {n+1}{n^2}
$$
The goal is to find the convergence rate. So we know,
$\lim\limits_{n \to \infty} \frac{n+1}{n^2} = 0$
so $\alpha = 0$.
Then it follows if we choose $b_n = \frac 1n$
$|a_n - \alpha| = \frac {n+1}{n^2} \le \frac {n+n}{n^2} = 2\frac{1}{n} = O(\frac 1n)$
I get lost at the point where they do $ \frac {n+n}{n^2} $. what are they doing to magnify the inequality here? I understand it's obvious the inequality still holds, but are they just choosing a polynomial that works out nicely relative to $b_n$ or is there a method to this? Sorry if this is trivial, but the numerical analysis text I have is awful and leaves out key explanations of why they did what they did - assuming it's obvious to the reader.
Thanks!

REPLY [0 votes]: To choose the $1/n$, a more intuitive method is to observe that $a_n=\frac{n(1+\frac{1}{n})}{n^2}=\frac{n(1+o(1)}{n^2}=\frac{1+o(1)}{n}\sim\frac{1}{n}$.


TITLE: Does $(I-BB^\dagger)C=0$ hold for a symmetric positive semidefinite matrix $G = \begin{pmatrix} A & B \\ B^T &C \end{pmatrix}$ with $C \preceq B$?
QUESTION [0 upvotes]: Given a symmetric positive semidefinite matrix
$$ G = \begin{pmatrix} A & B \\ B^T &C \end{pmatrix}$$
where $A$, $B$ and $C$ are not invertible, and $C\preceq B$, does the following equality hold?
$$ \left( I - B B^\dagger \right) C = 0 $$
Or, equivalently, does the following hold?
$$ C = B B^\dagger C$$
where $B^\dagger$ is the Moore-Penrose pseudoinverse of $B$.
I already know from here (Theorem 4.3) that we have
$$(I-AA^\dagger)B=0 \text{ and } (I-CC^\dagger)B^T=0$$ (we have it even without $C \preceq B$).
Edit : user 1551 proposed a direct counter-example to my proposition, I realized that it missed an assumption from my application: $C \preceq B$.

REPLY [0 votes]: Since $G\succeq0$, if $Cx=0$, then
$$
x^TBABx+2tx^TB^2x
=\pmatrix{x^TB&tx^T}\pmatrix{A&B\\ B&C}\pmatrix{Bx\\ tx}\ge0
$$
for every real number $t$. Therefore $\|Bx\|_2^2=x^TB^2x$ must be zero and in turn $Bx=0$. Hence $\ker(C)\subseteq\ker(B)$.
On the other hand, if $Bx=0$, then $-x^TCx=x^T(B-C)x\ge0$. Hence $Cx=0$ and $\ker(B)\subseteq\ker(C)$.
Thus $\ker(C)=\ker(B)$ and the orthogonal projections onto them coincide. That is, $I-CC^+=I-BB^+$. It follows that $(I-BB^+)C=(I-CC^+)C=0$.


TITLE: What is the intuition behind the uniqueness constraint for morphisms determining products?
QUESTION [3 upvotes]: I am following Bartosz Milewski's book and weblectures and have a question about the definition of the categorical product.
He uses names that are a bit different from the ones I can find on Google, so here is the definition. An object $c$ is the product of objects $a$ and $b$ (just denoted $c$, not $a \times b$) if it has the projections $p : c \rightarrow a$ and $q : c \rightarrow b$ such that for any object $c'$ together with projections $p' : c' \rightarrow a$ and $q' : c' \rightarrow b$ there exists an unique morphism $m : c' \rightarrow c$ that make the triangle commute, i.e. such that $p \cdot m = p'$ and $q \cdot m = q'$, i.e. "there exists only one of them that makes the triangles commute".
Now, what is the intuition behind this uniqueness constraint? I'm not looking for a counterexample, but for the "reason" behind it. What happens when we drop the uniqueness constraint? I know that if we do we satisfy all requirements for $a = a, b = b, c = a \times a \times b, c' = a \times b$, which we should not be able to. When we drop the uniqueness constraint, we allow for the situation that the morphism $m$ "makes up" something by selecting some random value for the unused $a$.
Can we state something about this in terms of bounds or entropy? I.e. something like 'a product is the "lowest-information situation" possible while still having all information available to be general enough to always satisfy the laws' or something like that? Like some greatest lower bound? An by enforcing $m$ to be unique we ensure that it cannot "add entropy/information" in any way?
Edit: I guess what I'm asking is: what is the consequence of one object having an unique mapping to another? If a morphism $a \rightarrow b$ is unique, what does this tell us about the information content of $a$ and $b$? Surely $a$ is "bigger"/contains more information than $b$, right? Doesn't $b$ even have to be a singleton in this sitution?
Edit2: OK let me reformulate my question. When we look at the object that forms the product, we are looking at the /smallest possible object/ that still satisfies all laws. Any bigger object we can "shave down" to this most basic object, and still retrieve the two components. Dually, when defining a sum, we want the most general object possible, any instantiation we can generalize. We define these greatest lower bounds and least upper bounds trough saying there must be an unique morphism between them. My question: how does the uniqueness of a morphism say anything about the size of the objects at its beginning and end?

REPLY [3 votes]: I want to address your edits first. A priori, categories have no notion of size, entropy, or information. You would need to formulate such a thing.
One possibility is by considering presheaves and the Yoneda embedding. I realize that these concepts will likely be unfamiliar to you right now, but they may help future visitors, including potentially yourself.
First reason for why we define products this way
Because this is the property that the Cartesian product of sets satisfies.
Brief review of presheaves and the Yoneda embedding
A presheaf on a category $\newcommand\C{\mathcal{C}}\newcommand\op{\text{op}}\C$ is a functor $\C^\op\to \newcommand\Set{\mathbf{Set}}\Set$.
The category of presheaves on $\C$ has presheaves for objects and natural transformations as morphisms, and is denoted $[\C^\op,\Set]$.
There is a canonical fully-faithful functor $y:\C\to [\C^\op,\Set]$ defined by
$y(x) = \C(-,x)$.
We can therefore view $y(x)$ as translating an abstract object into a collection of sets that we can measure the size of, and thus regard this as a sort of measure of the information contained in $x$.
Back to products
If $c$ has morphisms $p:c\to a$, and $q:c\to b$, then we can define a morphism
$y(c)\to y(a)\times y(b)$, where $y(a)\times y(b)$ means the pointwise cartesian product of sets (which is also the categorical product of the functors).
The morphism is defined as follows. Given $f\in y(c)(x) = \C(x,c)$,
$p\circ f : x\to a$ and $q\circ f : x\to b$, so defining
$f\mapsto (pf,qf)$ makes sense, and you can check that this is in fact natural.
Call this morphism $\phi_{p,q} : y(c)\to y(a)\times y(b)$.
The existence requirement translates into the statement that $\phi_{p,q}$ is (pointwise) surjective, and the uniqueness requirement translates into the statement that $\phi_{p,q}$ is (pointwise) injective. The two together become the statement that $\phi_{p,q}$ is an isomorphism. In a metaphorical sense, $c$ encodes the cartesian product of the information of $a$ and $b$.


TITLE: If Coulomb's were to depend on $\frac{1}{r^n}$ instead of $\frac{1}{r^2}$ where $n\neq 2$,would Gauss's law still be the same?
QUESTION [0 upvotes]: What I think is that Gauss's law is an independent law of nature which shouldn't change on modifying Coulomb's law.Am I correct?
Well the relation between Gauss's law and Coulomb's law is very strong but I can't strictly say that they can be derived from each other just like I can't say that law of conservation of linear momentum can be derived from Newton's 3rd law.
Edit
It's my gentle request that if one of them is a theorem then please name one of them as a theorem as it makes them quite complicated.

REPLY [3 votes]: No, Coulomb's law can be derived directly from Gauss law in three dimensions and vice-versa (assuming electric fields are governed by linear equations). Changing one implies that the other changes too.
However, if one measured the Coulomb's law to go as $\frac{1}{r^n}$ this could be a direct consequence of the world being $(n+1)$ dimensional with the Gauss law intact.


TITLE: Probability that the sum of two integers randomly picked from a range of [0;150] will be less or equal to 256?
QUESTION [0 upvotes]: What is the probability that the sum of two integers randomly picked from a range of $[0;150]$ will be less or equal to 256? Integers can be the same. This includes 0 and 150 and must be a real integer

REPLY [1 votes]: Are you okay with the idea of adding conditional probalities with conditions are indepent.
For example: if there is a 75% chance of rain and if it rains there is a 10% chance I'll take a walk.  And if it doesn't rain there is a 60% chance I'll take a walk. Then the chance that it rains and I take a walk is the probability of rain times the probability of a walk give it rains.  ($0.75 \times 0.1 = 0.075$). And the probability of it doesn't rain and I take a walk is the probabily of no rain, times the probability of a walk given no rain. ($0.25\times 0.6 = 0.15$).  As the event of rain and not rain are mutually exclusive we can add those probabilities.  ($0.075 + 0.15 = 0.225$) so the probability that I take a walk is 22.5%.
Are you okay with that.
So if for your first number, $a$ you pick anything equal to or less than $a \le 256-150 = 106$ then the probability that your second number will make a sum less or equal to $256$ is a certainty; $100\%$.  That is if $a \le 106$ and $a + b \le a + 150 \le 106 + 150 = 256$.  That is a certainty.
So the probability of the first being less or equal to $106$ is: There are $151$ choices for the first number ($0$ through $150$); and there are $107$ ways the number is $\le 106$ (it may be $0... 106$.) So the probability of your first number being less or equal to $106$ is $\frac {107}{151}$.  And IF that occured to probability of you second number being $150$ or less to make the sum $256$ or less is $1$.  It must happen.
So the probability that 1) your first number is $\le 106$ and 2) the sum of your numbers is $\le 256$ is:  $\frac {107}{151} \times 1=\frac {107}{151}$.
Now if instead you first number was $a$ and $a > 106$.  The for you sum $a +b \le 256$, we muat have $b \le 256 -a$.  So $b$ can be anything from $0$ to $256 -a$.  There are $257-a$ possible ways this can happen so IF your first number is $a$ then the probability that the second number, $b$, is $\le 256-a$ is $\frac {257-a}{151}$.
And the probability that the first number is $a$ is $\frac 1{151}$.  So the probability that your first number is $a$ and your second number is $\le 256 a$ is: $\frac 1{151}\cdot \frac {257-a}{151}$.
So if you add all thes probabilities toger you get:
Probability sum is $256$ or less $= \frac {107}{151} + \sum\limits_{a=107}^{150}  \frac 1{151}\frac{257-a}{151}=$
$\frac {107}{151} + \frac 1{151^2}\sum\limits_{a=107}^{150} (257-a)$
So what is $\sum\limits_{a=107}^{150} (257-a)$?
Well it is $\sum_{a=107}^{150}257 + \sum_{a=107}^{150} a$.
And $\sum_{a=107}^{150} 257 = 257 + 257 +...+257 = 257(150-106) = 257*44$.
And $\sum_{a=107}^{150} a = (1 + 2+ 3 +....... + 150) - (1+2+3+...... + 106)= $
$\frac {150*151}2 - \frac {106*107}2 = 75*151 - 53*107$.
So $\sum\limits_{a=107}^{150} (257-a) = 257*44- 75*151 + 53*107$
And the probality we want is:
$\frac {107}{151} + \frac 1{151^2}(257*44 - 75*151 + 53*107)=$
$\frac {107 + \frac {257*44+53*107}{151} - 75}{151}=$
$\frac {32 +\frac {257*44+53*107}{151} }{151}=$
$\frac {144\frac {67}{151}}{151} \approx 95.7\%$.


TITLE: Bochner integrability within a subspace
QUESTION [1 upvotes]: Let $(H,||\cdot||_H)$ be a Banach space and $K$ a (not necessarily closed) subspace. Suppose that $K$ is a Banach space under another norm $||\cdot||_K$, which satisfies
$$||x||_H\leq ||x||_K$$
for all $x\in K$. Let $(S,\mu)$ be a measure space and $f:S\rightarrow K$ a strongly measurable function in the sense of Bochner, with respect to both $||\cdot||_K$ and $||\cdot||_H$. That is, there exist sequences of simple functions $\phi_n$ and $\psi_n$ on $S$ taking values in $K$ and $H$ respectively such that
$$\lim_{n\rightarrow\infty} ||f(s) - \phi_n(s)||_{K} = \lim_{n\rightarrow\infty} ||f(s) - \psi_n(s)||_{H} = 0$$
for $\mu$-almost all $s\in S$.
Suppose now that $f$ satisfies
$$\int_S ||f(s)||_H\,d\mu(s) < \infty,$$
so that, by a criterion of Bochner, $f$ is integrable as a function with values in $H$. Denote
$$h := \int_S f(s) d\mu(s).$$
Question: If $h\in K$, then does it follow that $f$ is Bochner-integrable in $K$? That is, does there exist a sequence of simple functions $\phi_n$ such that
$$\lim_{n\rightarrow\infty}\int_S ||f(s) - \phi_n(s)||_{K}\,d\mu(s) = 0?$$

REPLY [3 votes]: The Bochner integral is a red herring here.  Let $S = \mathbb{N}$ and let $\mu$ be counting measure.  Then a "strongly measurable function" is just a sequence, and it is "Bochner integrable" iff it is absolutely summable (in $H,K$ respectively).
If the $H$ and $K$ norms are not equivalent (which is necessarily the case when $K$ is not closed) then we can find a sequence $y_n \in K$ with $\|y_n\|_H \le 2^{-n}$ but $\|y_n\|_K \ge 2^n$.  Set $x_1 = y_1$, $x_{n+1} = y_{n+1} - y_n$, and think of the function $f(n) = x_n$.  Note that $\|x_n\|_H \le 2^{-(n-1)}$ and $\|x_n\|_K \ge 2^{n-1}$.
Now $x_n$ is absolutely summable in $H$-norm and $\sum_n x_k = 0$ because the sum telescopes.  That is to say that $f$ is Bochner integrable in $H$ and the value of the integral $\int_S f\,d\mu$ is 0 which is in $K$.
On the other hand, $\sum_n x_n$ diverges in $K$, which is to say that the Bochner integral in $K$ does not exist.


TITLE: Block matrix characteristic polynomial
QUESTION [0 upvotes]: Let $n \in \Bbb N^*$ and $A \in \cal M_n(\Bbb R)$ a square matrix.  Let the block matrix
$$B = \begin{pmatrix}A&A\\A&A \end{pmatrix} \in \cal M_{2n}(\Bbb R)$$

Calculate the  characteristic polynomial $\chi_B$ using $\chi_A$.
Proof that $A$ is diagonalisable $\iff$ $B$ is diagonalisable.

REPLY [2 votes]: For the first part: we have
$$
tI_{2n} - B =
\pmatrix{tI_n - A & -A\\ -A & tI_n - A}
$$
Noting that these matrices commute (or in particular, that the lower two matrices commute), we have
$$
\begin{align}
\det(tI_{2n} - B) &= \det((tI_n - A)^2-A^2)\\
&= \det[((tI_n - A)-A)((tI_n - A)+A)]\\
&= \det((tI_n - A)-A) \det((tI_n - A)+A)\\
&= \det((tI_n - 2A) \det(tI_n)\\
&= \det(2((t/2) - A)) \det(tI_n)\\
&= 2^n \det((t/2) - A) t^n\\
&= 2^n t^n \chi_A\left(\frac t2\right)
\end{align}
$$
Second part: I haven't put it all together nicely, but here are some potentially helpful thoughts
Thought 1: If $B$ is diagonalizable, then $B$ has $2n$ linearly independent eigenvectors.  Now, let
$$
v = \pmatrix{v_1\\v_2}
$$
be an eigenvector of $B$, with $v_1,v_2 \in \mathbb{C}^n$.  What can we say about the vectors $v_1$ and $v_2$?  Remember that $B$ has the eigenvector $0$ of multiplicity $n + \operatorname{null}(A)$.
Thought 2:
$$
\pmatrix{I & I\\ I& -I} \pmatrix{A & A\\ A & A}\pmatrix{I& I\\I & -I} =
\pmatrix{4A & 0\\0 & 0}
$$
Or, in particular,
$$
\frac{1}{2}\pmatrix{I & I\\ I& -I} \pmatrix{A & A\\ A & A}
\pmatrix{I& I\\I & -I} = \\
\left(\frac{1}{\sqrt{2}}\pmatrix{I & I\\ I& -I}\right)^{-1} \pmatrix{A & A\\ A & A} \frac{1}{\sqrt{2}}\pmatrix{I& I\\I & -I} =
\pmatrix{2A & 0\\0 & 0}
$$


TITLE: The underlying space of a scheme remembers its affineness?
QUESTION [6 upvotes]: Let $f:X\rightarrow Y$ be a morphism of schemes. We know that if $Y$ is affine and $f$ induces homeomorphism on the underlying spaces then $X$ is affine. Is it true that if $X$ is affine and $f$ induces a homeomorphism on the underlying spaces then $Y$ is affine?
More generally, is it true that a scheme whose underlying space is homeomorphic (possibly via a homeomorphism that is not induced by a morphism of schemes) to the underlying space of an affine scheme is affine? EDIT: actually, the answer to the last question is a very strong "NO" as the underlying space of any scheme is sober and Noetherian sober spaces are spectral (i.e. homeomorphic to the underlying space of an affine scheme).

REPLY [5 votes]: As pointed out by abx here, the following example fits the bill.
Take a nodal cubic in $\mathbb{P}^2$, this is an irreducible scheme proper over $\mathbb{C}$, take its normalization and throw out a point from the inverse image of the node. We get a $\mathbb{C}$-morphism from the affine line to our cubic, which is bijective on the underlying spaces. Since a non-empty proper closed subspace of either the affine line or the cubic is a finite union of closed points, this morphism is also closed, thus it induces a homeomorphism on the underlying spaces. I find it somewhat amusing that this example was not pointed out earlier (indeed, it seems to be even simpler than Julian Rosen's much upvoted example).


text:
\begin{document}

\def \N {\mathbb{N}}
\def \Z {\mathbb{Z}}
\def \Q {\mathbb{Q}}
\def \R {\mathbb{R}}
\def \C {\mathbb{C}}
\def \Cbar {\overline{\mathbb{C}}}
\def \D {\mathbb{D}}
\def \Dr {\mathbb{D}_{r}}
\def \L {\mathcal{L}}
\def \M {\mathcal{M}}
\def \T {\mathcal{T}}
\def \O {\mathcal{O}}
\def \F {\mathcal{F}}
\def \zbar {\overline{z}}
\def \eps {\varepsilon}
\def \epsg {\varepsilon > 0}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}{Proposition}[section]

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}

\title{A note on the run length function for intermittent maps}

\author{{\bf\large Hongfei Cui}\hspace{2mm}\vspace{1mm}\\

{\it\small Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences}, {\it\small Wuhan  430071,  China}\\
{\it\small E-mail:  cui@wipm.ac.cn}\vspace{1mm}\\
{\bf\large Lulu Fang}\\%\footnote{Corresponding author.}

{\it\small School of Mathematics},\\ {\it\small Sun Yat-sen university}, {\it\small Guangzhou  510275,  China}\\
{\it\small E-mail:  fanglulu1230@163.com}\vspace{1mm}\\
{\bf\large Yiwei Zhang}\footnote{Corresponding author, and every author contributes equally.}\hspace{2mm}\vspace{1mm}\\
{\it\small School of Mathematics and Statistics, Center for Mathematical Sciences},\\
{\it\small Hubei Key Laboratory of Engineering Modeling and Scientific Computing},\\
{\it\small Huazhong University of Sciences and Technology},
{\it\small Wuhan 430074,  China}\\
{\it\small E-mail: yiweizhang@hust.edu.cn}\vspace{1mm}}

\date{\today }
\maketitle

\begin{abstract}
We study the run length function for intermittent maps. In particular, we show that the longest consecutive zero digits (resp. one digits) having a time window of polynomial (resp. logarithmic) length. Our proof is relatively elementary in the sense that it only relies on the classical Borel-Cantelli lemma and the polynomial decay of intermittent maps. Our results are compensational to the Erd\H{o}s-R\'{e}nyi law obtained by Denker and Nicol in \cite{dennic13}.
\end{abstract}

\bigskip

\section{Introduction}

Consider a piecewise monotone interval map $T:[0,1)\to[0,1)$ with a countable or finite partition $\{I_{j}\}$, preserving a probability measure $\nu$. Given an $x\in[0,1)$ and $k\in\N$, set $\varepsilon_{k}(x)=j$ if $T^{k-1}(x)\in I_{j}$ for some $j$.
The \emph{run length function} for this system is defined as the maximal length of consecutive $j$ digits in the sequence $(\varepsilon_{1}(x),\cdots, \varepsilon_{n}(x))$. Namely,
$$
r_n(x,j)=\max\{k\geq 0:\varepsilon_{i+1}(x)=\cdots=\varepsilon_{i+k}(x)=j~~~\mbox{for some}~~0\leq i\leq n-k\}.
$$

In this note, we concern about the run length function for a class of intermittent maps, and show the existence of an appropriate scaling length $k(n)$, to quantify the asymptomatic behavior of $r_{n}(x,j)$ for $\nu$-typical $x$.

The studies on the run length functions are motivated from both theoretical and practical aspects. First, it has been revealed that run length function of piecewise linear interval maps has several probabilistic interpretations, which have great applications in DNA string machine \cite{AGW90}, reliability theory and non-parametric statistics (see e.g., \cite{Bal01,Bar92,Bat48,Mus00,Nov01,Rev90} and the references therein). For example, when specializing $T(x) = 2x({\rm mod}~1)$ and $\nu$ being the Lebesgue measure on $[0,1)$, the unit interval endows with a finite partition $\{I_0,I_1\}$, where $I_0 =[0,1/2)$ and $I_1=[1/2,1)$. The run length function for such a system corresponds to the longest length of consecutive terms of ``heads" in a mathematical experiment of coin tossing (originally back to de Moivre Problem LXXIV in the year of 1738 \cite{deMo1738}). Indeed, Erd\H{o}s and R\'{e}nyi \cite{Eroren70} obtained that for Lebesgue almost surely $x \in [0,1)$, both $r_n(x,0)=r_n(x, 1)$ increase to infinity with the logarithmic speed $\log_2 n$ as $n$ goes to infinity.

The above discussions of run length function has been generalized into $\beta$-transformations ($\beta>1$)\cite{Tonyuzhao}, that is, $T(x) = \beta x({\rm mod}~1)$ and $\nu$ is the Parry measure on $[0,1)$ (see \cite{Par60}). In this case, the unit interval endows with a finite partition $I_0 = [0,1/\beta),I_1 = [1/\beta,2/\beta), \cdots, I_{\lceil\beta\rceil-1} = [(\lceil\beta\rceil-1)/\beta,1)$, where $\lceil\cdot\rceil$ denotes the smallest integer no less than some number. As an important by-product, the length of cylinder can be partially estimated in terms of run length function. This fact is critical in understanding many dynamical problems of $\beta$-transformations (see \cite {FW12, FWL16}). Unfortunately, comparing to $\beta$-transformations, relatively limited work has been done in studying the run length function for interval maps with non-linear components. This forms the first goal of the note.

\medskip

Besides, the run length function also associates with the so-called Erd\H{o}s-R\'{e}nyi law. Given an observation $\varphi$ in $L^{1}(\nu)$, denote by $S_{n}(\varphi):=\sum_{i=1}^{n-1}\varphi\circ T^{i}$ and
\begin{align}\label{equ_edos-renyi}
 \theta(n,K(n),\varphi)&:=\max_{0\leq i\leq n-K(n)}\{S_{i+K(n)}(\varphi)-S_{i}(\varphi)\}\nonumber\\
 &=\max\{S_{K(n)}(\varphi)\circ T^{i}:0\leq i\leq n-K(n)\},~~~~~~\forall n\in\N,
\end{align}
i.e., the maximal Birkhoff sum gaining over a window of length $K(n)$ up to time $n$. The \emph{Erd\H{o}s-R\'{e}nyi law} concerns about the existence of an appropriate scaling length $K(n)$, such that the asymptotic behavior of the fraction $\frac{\theta(n,K(n),\varphi)}{K(n)}$ has a non-degenerate limit for $\nu$-typical $x$. Inspired by the work of Erd\H{o}s and R\'{e}nyi for i.i.d random valuables \cite{Eroren70}, it is believed that under certain hyperbolicity of $T$ and regularity of $\varphi$ hypothesis, the behavior of $\theta(n,K(n),\varphi)$ asymptotically admit a dichotomy on exhibiting either ordinary law of large numbers or asymptotic law of long length function, subject to the scaling length of $K(n)$.

Initialized by the work of Grigull \cite{Gri93} for hyperbolic rational maps, there are a number of results on the studies of Erd\H{o}s-R\'{e}nyi law(as well as answering the above dichotomy), particularly for the dynamical systems satisfies the large deviation principle. For examples, see the work of Chazottes and Collet \cite{Chacol01} for uniformly expanding interval maps, and the work of Denker and Kabluchko \cite{denzab07} for Gibbs-Markov dynamics.

Later, Denker and Nicol \cite{dennic13} provide several extensions of Erd\H{o}s-R\'{e}nyi law to the non-uniformly expanding dynamical systems, including logistic-like maps and intermittent maps. Unfortunately, if the dynamical systems are absent of large derivation principle, only partial results can be obtained, and the regularity of the observation function $\varphi$ has to be assumed Lipschitz/H\"{o}lder continuous (rather than integrable). Under this framework, the characteristic functions are inapplicable to Denker and Nicol's result, and thus no results on the run length function have been known. Moreover, their proofs (e.g., \cite[Theo 4.1]{dennic13} for the intermittent maps) heavily rely on the previous dynamical Borel-Cantelli lemma results obtained Gouez\"{e}l \cite[Theo 1.1]{lesGou07}, and large deviation results obtained by Pollicott and Sharp \cite[Theo 4]{Polsha09}. Our second goal is aiming to fulfill this gap of the run length function for the intermittent maps with a proof in a more elementary way (namely, without using the two technical machineries above). This will in our belief provide a more refined description in this research direction.

\medskip

Let us now state our results more mathematically. Consider $0<\alpha<1$ and recall the \emph{intermittent maps} $T_{\alpha}: [0,1) \to [0,1)$ is defined as
\[
T_{\alpha}(x)=
\begin{cases}
x(1+2^\alpha x^\alpha),  &\text{if $0\leq x <1/2$};\\
2x-1,  & \text{if $1/2 \leq x <1$}.
\end{cases}
\]
It is well known that $T:=T_{\alpha}$ has a finite absolutely continuous invariant probability measure $\mu$.  Let $I_0 = [0,1/2)$ and $I_1 = [1/2,1)$, and we are accordingly concerned with the length of the longest consecutive zero digits in $\left( \varepsilon_1(x), \varepsilon_2(x), \cdots, \varepsilon_n(x) \right)$, and the length of the longest consecutive one digits for $\mu$-almost all $x \in [0, 1)$. Set
\[
r^\alpha_n(x)= \max\Big\{k \geq 0: \varepsilon_{i+1}(x)=\cdots=\varepsilon_{i+k}(x)=0\ \text{for some}\ 0 \leq i\leq n-k\Big\},
\]
and
\[
R^\alpha_n(x)= \max\Big\{k \geq 0: \varepsilon_{i+1}(x)=\cdots=\varepsilon_{i+k}(x)=1\ \text{for some}\ 0 \leq i\leq n-k\Big\}.
\]
With this conventions, our main theorem is stated as follows.
\begin{theorem}
For $\mu$-almost all $x \in [0,1)$,
\begin{equation}\label{equ_polynimal}
\lim_{n \to \infty} \frac{\log r^\alpha_n(x)}{\log n^{\alpha}} =1,
  \end{equation}
  and
 \begin{equation}\label{equ_exponentially}
\lim_{n \to \infty} \frac{R^\alpha_n(x)}{\log_2n} =1.
  \end{equation}
\end{theorem}

Recall that $S_n(\chi_{I_0}(x))$ is the number of positive numbers  $0 \leq i \leq n-1$ such that $T^i(x) \in I_0$, i.e.,
$$
S_n(\chi_{I_0}(x)) = \sum_{i=0}^{n-1} \chi_{I_0}\circ T^i(x),
$$
where $\chi_{I_0}$ is the indicator function of the interval $I_0$. One can easily obtain the following corollary, which says that for a particular observable the limit of the maximal average over a time window of the length $n^{\alpha_1}$ (with $\alpha_1 <\alpha$) is equal to $1$ $\mu$-almost surely.

\begin{corollary}
If $\alpha_1<\alpha$, then for $\mu$-almost $x \in [0, 1)$,
$$
\lim_{n\to \infty} \frac{\max \{ S_{n^{\alpha_1}} (\chi_{I_0} \circ T^i(x)) :  \ 0\leq i\leq n-n^{\alpha_1} \} }  {n^{\alpha_1}}= 1.
$$
\end{corollary}
Similarly, we can obtain
\begin{corollary}
Let $S_n(\chi_{I_1}(x)) = \sum_{i=1}^{n} \chi_{I_1}\circ T^i(x)$.
Then for any integer sequence  $k(n)$ that satisfying $\limsup_{n\to \infty} \frac{k(n)}{\log_2 n} \leq 1$, and for $\mu$-almost $x \in [0, 1)$, we have
$$
\lim_{n\to \infty} \frac{\max \{ S_{k(n)} (\chi_{I_1} \circ T^i(x)) :  \ 0\leq i\leq n-k(n) \} }  {k(n)}= 1.
$$
\end{corollary}

The theorem indicates that polynomial length consecutive digits is typical for the run length function of intermittent maps, and different order of the run length function can coexist (differing from the phenomenon in $\beta$-transformations). The result implies the \cite[Theo 4.1(b)]{dennic13} for $\phi =\chi_{I_0}$ a indicator observable of $I_0$.

Our proof relies heavily  on the classical Borel-Cantelli lemma and the polynomial decay of intermittent maps. We obtain the upper limit by using the equivalence between the absolutely continuous invariant measure $\mu$ and Lebesgue measure outside a small one-side neighborhood of $0$. To get a result of other direction, we break the first $n$ iterations into disjoint blocks with a small scale, on which the digits are not  all $0$ or $1$. Then using the correlations between different blocks, and combining the measure estimation of the consecutive $0$ or $1$ digits, we obtain the lower limit of the run length function.

\section{Proof of theorem}

Note that $T_{\alpha}$ has a finite absolutely continuous invariant probability measure $\mu$
with a density function $h(x) = d\mu/dx$ satisfying
\[
\lim_{x \to 0} x^\alpha h(x) =c
\]
for some constant $c$ (see Theorem A of Hu \cite{lesHu04}). As a consequence, we obtain there exists a constant $1<C<\infty$ such that
\begin{equation}\label{mu}
C^{-1}x^{1-\alpha} \leq \mu([0,x)) \leq Cx^{1-\alpha}
\end{equation}
for sufficiently small $x \in [0,1)$. Moreover, $C \leq h(x) \leq C x^{1-\alpha}$ for $x \in [a,1)$ with $a>0$, then the measure $\mu$ and Lebesgue measure are uniform equivalent on every interval $[a, 1)$, see \cite{Lsv99, lesYoung}. It follows that there exists a constant $\tilde{C}< \infty$ such that for any interval $A \subset [0,1)$, we have
\begin{equation}\label{mu0}
\frac{1}{\tilde{C}} |[1/2, 1) \cap A|\leq \mu( [1/2, 1) \cap A) \leq \tilde{C}|A|.
\end{equation}

We now turn to the proof of the theorem.
\begin{proof}
The proof is divided into three  parts:
\\
{\bf Part I.} We will prove that for $\mu$-almost all $x \in [0,1)$,
\[
\limsup_{n \to \infty} \frac{\log r^\alpha_n(x)}{\log n} \leq \alpha.
\]
Let $0<\varepsilon<1-\alpha$. It suffices to  show that $\mu \{x\in [0,1): r^\alpha_n(x) \geq n^{\alpha+\varepsilon}\ i.o.\} =0$. In fact, if $r^\alpha_n(x) \geq n^{\alpha+\varepsilon}$ holds for some $x \in [0,1)$ and infinitely many $n \in \mathbb{N}$, then we obtain two cases that either $\varepsilon_{i}(x)=0$ for all $i$, $1\leq i \leq \lceil n^{\alpha+\varepsilon}\rceil$ and infinitely many $n$,   or there exists $1\leq i\leq n-\lceil n^{\alpha+\varepsilon}\rceil$ such that
\[
\varepsilon_{i}(x)=1 \ \ \text{and} \  \varepsilon_{i+1}(x)= \cdots =\varepsilon_{i+\lceil n^{\alpha+\varepsilon}\rceil}(x)= 0 \ \ \text{for infinitely many} \ n.
\]
Put $a_0=1/2$ and $a_{n+1}=T^{-1}(a_n) \cap [0, 1/2)$. If $\varepsilon_{i}(x)=0$ for all  $1\leq i \leq \lceil n^{\alpha+\varepsilon}\rceil$, then it follows from Lemma 3.2 of \cite{Lsv99} that
\[
x\leq a_{\lceil n^{\alpha+\varepsilon}\rceil} \leq C\frac{1}{\lceil n^{1+\frac{\varepsilon}{\alpha}}\rceil}.
\]
Therefore, we obtain
\begin{equation}\label{mu1}
\mu\{x: \ \varepsilon_{i}(x)=0 \ \text{for all}  \ 0\leq i \leq \lceil n^{\alpha+\varepsilon}\rceil \ \ i.o. \}=0.
\end{equation}

For the second case, it means that there exist $1\leq i \leq n-\lceil n^{\alpha+\varepsilon}\rceil$ and infinitely many $n$ such that
 \[
 T^{i-1}(x)\in I_1 \cap T^{-1}[0, a_{\lceil n^{\alpha+\varepsilon}\rceil}] \subset I_1 \cap T^{-1}[0, a_{\lceil i^{\alpha+\varepsilon}\rceil}].
 \]

Note that $n^{\alpha+\varepsilon}$ is increasing to infinity as $n$ goes to infinity. So, we obtain that either $\varepsilon_{i}(x)=1$ and $\varepsilon_j(x)=0$ for all $j>i$, or $T^{i-1}(x) \in I_1 \cap T^{-1}[0, a_{\lceil i^{\alpha+\varepsilon}\rceil}]$ for infinitely many $i$. It is easy to show that
\begin{equation}\label{mu2}
\mu\{x: \varepsilon_{i}(x)=1 \ \text{and} \ \varepsilon_{j}(x)=0 \ \text{for all} \ j\geq i \}=0.
\end{equation}
Since
\begin{align*}
&\sum_{i=1}^{\infty} \mu\{x:\  \ T^{i-1}(x) \in I_1 \cap T^{-1}[0, a_{\lceil i^{\alpha+\varepsilon}\rceil}] \ \} \\
&=\sum_{i=1}^{\infty} \mu\{I_1 \cap T^{-1}[0, a_{\lceil i^{\alpha+\varepsilon}\rceil}]\}\leq \sum_{i=1}^{\infty} \frac{\tilde{C}}{2i^{1+\frac{\varepsilon}{\alpha}}} < \infty,
\end{align*}
 by Borel-Cantelli lemma, it follows that
\begin{equation}\label{mu3}
\mu \{x: T^{i-1}(x) \in I_1 \cap T^{-1}[0, a_{\lceil i^{\alpha+\varepsilon}\rceil}] \  \text{ for infinitely many} \ i \}=0.
\end{equation}
Combining (\ref{mu1}, \ref{mu2}, \ref{mu3}), we obtain
\[
\mu\{x\in [0,1): r_n^{\alpha}(x)\geq n^{\alpha+\varepsilon} \ i.o. \}=0.
 \]
 Therefore, for $\mu$-almost all $x\in [0,1)$,
 \[
 \limsup_{n\to \infty} \frac{\log r_n^{\alpha}(x)}{\log n} \leq \alpha+\varepsilon.
 \]
 Letting $\varepsilon \to 0$, we complete the proof of Part I.

{\bf Part II.} We will prove that
for $\mu$-almost all $x \in [0,1)$,
\[
\liminf_{n \to \infty} \frac{\log r^\alpha_n(x)}{\log n} \geq \alpha.
\]

We  shall use the following statement. Let $\xi=\{I_0, I_1 \}$ be a partition of $[0, 1)$ and $\xi_n= \xi \bigvee T^{-1}\xi\bigvee \dots \bigvee T^{-n+1}\xi$, where $\xi \bigvee \eta=\{A \cap B: A\in \xi, B\in \eta\}$. In \cite{lesHu04}, Hu showed that there exists $C_1>0$ and $l>0$ such that for any $m\geq 0$ and $A\in\xi_m$ and for any measurable set $B\subset [0, 1) $,
\begin{equation}\label{equ:polynimal}
|\mu(A \cap T^{-n-m}B) - \mu(A) \mu(B)| \leq \frac{C_1 m^{1/\alpha-1}}{(n-l)^{1/\alpha -1}} \mu(A) \mu(B), \  \text{for}\ n>l.
\end{equation}
Then by the invariance of $\mu$, we obtain
\[
|\mu(A^{c} \cap T^{-n-m}B) - \mu(A^c) \mu(B)| \leq \frac{C_1 m^{1/\alpha-1}}{(n-l)^{1/\alpha -1}} \mu(A) \mu(B),
\]
where $A^{c}:=[0, 1) \setminus A$. It implies that
\begin{equation}\label{hu}
\mu(A^{c} \cap T^{-n-m}B) \leq \left( \frac{C_1 m^{1/\alpha-1}}{(n-l)^{1/\alpha -1}} \mu(A) + \mu(A^c)   \right) \mu(B).
\end{equation}

For any $x \in [0,1)$ and $m,n \in \mathbb{N}$ with $m<n$, we define
\[
r^\alpha_{m,n}(x):= \max\{k \geq 0: \varepsilon_{i+1}(x)=\cdots= \varepsilon_{i+k}(x)=0\ \text{for some}\ m-1\leq i \leq n-k\}.
\]
Thus, $r^\alpha_{1,n} = r^\alpha_{n}$.
For any $0<\varepsilon<\alpha$, write $t_n:= \lceil n^{\alpha -\varepsilon}\rceil$ and $k_n: = \lfloor \frac{n}{t^{1+\varepsilon}_n}\rfloor$.
Let
\[
E = \{x\in [0,1): r^\alpha_{t_n}(x) < t_n\}
\]
and
\[
F_j =\{x \in [0,1): r^\alpha_{jt^{1+\varepsilon}_n+1, jt^{1+\varepsilon}_n+t_n}(x) < t_n\},\ \ 1 \leq j \leq k_n-1.
\]
Then
\begin{align*}
\{x \in [0,1): r^\alpha_n(x) < t_n\} &\subseteq E \bigcap F_1 \bigcap \cdots \bigcap F_{k_n-1}\\
&= E \bigcap T^{-t^{1+\varepsilon}_n}(E\cap F_1\cap \cdots \cap F_{k_n-2}).
\end{align*}

Let \[
I_{k}(0,\cdots,0):=\{x \in [0,1):\varepsilon_1(x) = \cdots = \varepsilon_k(x)=0\}\ \text{for some}\ k \geq 1.
\]
Then $I_{t_n}(0,\cdots,0) \in \xi_{t_n}$ and  ${I_{t_n}(0,\cdots,0)}= E^c$. By (\ref{mu}), the similar arguments in Section 6.2 of \cite{lesYoung} imply that
\[
\widetilde{C}^{-1} \frac{1}{t^{1/\alpha -1}_n}\leq \mu(I_{t_n}(0,\cdots,0)) \leq \widetilde{C}\frac{1}{t^{1/\alpha -1}_n},
\]
where $\widetilde{C}>1$ is a constant. For $n$ big enough so that $t_n^{1+\varepsilon} -t_n > l$,
combining this with (\ref{hu}), we deduce that
\begin{align}\label{1}
\mu\{x \in [0,1): r^\alpha_n(x) < t_n\} &\leq \mu(E \bigcap T^{-t^{1+\varepsilon}_n}(E\cap F_1\cap \cdots \cap F_{k_n-2})) \notag\\
&\leq \left(  \frac{C_1t_n^{1/\alpha-1 }}{(t_n^{1+\varepsilon}-t_n -l)^{1/\alpha-1}}\mu(I_{t_n}(0,\dots, 0))+ \mu(E) \right) \notag\\
&~~~~\times \mu (E\cap F_1\cap \cdots \cap F_{k_n-2}) \notag\\
& \leq \cdots \cdots \notag\\
&\leq \left(  \frac{C_1 \widetilde{C} }{(t_n^{1+\varepsilon}-t_n -l)^{1/\alpha-1}}  + 1-  \frac{\widetilde{C}^{-1} }{t^{1/\alpha -1}_n} \right)^{k_n-1}.
\end{align}

Using the inequality: $e^{-x} \geq 1-x$ for any $x \geq 0$, we have
\begin{align*}
\mu\{x \in [0,1): r^\alpha_n(x) < t_n\} & \leq \left(1- \frac{\widetilde{C}^{-1} }{t^{1/\alpha -1}_n} + o(\frac{1 }{t^{1/\alpha -1}_n}) \right)^{k_n-1}\\
&\leq  \exp\left( -\frac{\widetilde{C}^{-1} }{t^{1/\alpha -1}_n} \frac{n}{t^{1+\varepsilon}_n} \right)\\
&=\exp\left(-\widetilde{C}^{-1} n^{-(\frac{\varepsilon }{\alpha}+\varepsilon^2-\alpha\varepsilon)} \right).
\end{align*}

Since $\varepsilon/\alpha + \varepsilon^2 > \alpha \varepsilon$, we eventually get that
\[
\sum_{n \geq 1} \mu\{x \in [0,1): r^\alpha_n(x) < t_n\} <\infty.
\]
By Borel-Cantelli lemma, we conclude that
for $\mu$-almost all $x \in [0,1)$,
\[
\liminf_{n \to \infty} \frac{\log r^\alpha_n(x)}{\log n} \geq \alpha-\varepsilon.
\]
Letting $\varepsilon \to 0^+$, the proof of Part II is completed.

{\bf Part III.} We are concerned in this part with the run length function $R^\alpha_n(x)$.
This part can be proved by the similar way as shown before.
We first show that for $\mu$-almost all $x \in [0,1)$,
\[
\limsup_{n \to \infty} \frac{R^\alpha_n(x)}{\log_2n} \leq 1.
\]
In fact, let $\varepsilon>0$, if $R_n^{\alpha}(x) \geq (1+\varepsilon)\log_2n$ holds for some $x\in [0, 1)$ and $n\in \mathbb{N}$, then there exists $0\leq i \leq n-k$ (where $k=\lceil (1+\varepsilon) \log_2n \rceil$) such that
$$
\varepsilon_{i+1}(x)=\dots =\varepsilon_{i+k}(x)=1.
$$
That is, $T^i(x) \in I_1$, $T^{i+1}(x) \in I_1$, $\dots$, $T^{i+k-1}(x)\in I_1$. Hence
\[
1-\frac{1}{(i+1)^{1+\varepsilon}} \leq 1-\frac{1}{2^k} \leq T^i(x) < 1.
\]
Since $(1+\varepsilon) \log_2n \to \infty$ as $n \to \infty$, $R_n^{\alpha}(x) \geq (1+\varepsilon)\log_2n$ for infinite many $n$ implies that
either $T^i(x)\in [1-\frac{1}{{(i+1)}^{1+\varepsilon}}, 1]$ for infinite many $i$ or there exists an $i\in \mathbb{N}$ such that $\varepsilon_j=1$ for all $j\geq i$. Then it follows from Borel-Cantelli lemma that
\[
\mu\{x\in [0,1): R_n^{\alpha}(x) \geq (1+\varepsilon)\log_2n \ i.o.\ \}=0.
\]
Therefore, for $\mu$-almost all $x\in [0,1)$,
\[
\limsup_{n \to \infty} \frac{R_n^{\alpha}(x)}{\log_2n}\leq 1+\varepsilon.
\]
Taking $\varepsilon \to 0$, we obtain the proof of the first part.

Next we will show for $\mu$-almost all $x \in [0,1)$,
\[
\liminf_{n \to \infty} \frac{R^\alpha_n(x)}{\log_2n} \geq 1.
\]
We  shall use the following facts. For any interval $A \subseteq (1/2,1)$ and interval $B \subseteq (1/2,1)$,
\begin{equation*}
|\mu(A\cap T^{-n}B) - c_n\mu(A)\mu(B)| \leq \frac{1}{n^{1/\alpha}}\mu(B),\ \ \forall n\geq 1,
\end{equation*}
where $c_n =  1+ \frac{s}{n^{1/\alpha-1}} + o(\frac{1}{n^{1/\alpha-1}})$ for some non-zero constant $s$, see Eq. (1.3) of Gou\"{e}zel \cite{lesGou07}. Then
\begin{equation}\label{mixing}
\mu(A^c\cap T^{-n}B) \leq \left(c_n\mu(A^c)+ \frac{1}{n^{1/\alpha}}\right)\mu(B),\ \ \forall n\geq 1.
\end{equation}

For any $x \in [0,1)$ and $m,n \in \mathbb{N}$ with $m<n$, we define
\[
R^\alpha_{m,n}(x):= \max\{k \geq 0: \varepsilon_{i+1}(x)=\cdots= \varepsilon_{i+k}(x)=1\ \text{for some}\ m-1\leq i \leq n-k\}.
\]
Thus, $R^\alpha_{1,n} = R^\alpha_{n}$.
For $\varepsilon>0$ small such that $\frac{\alpha}{1-\alpha} - \varepsilon -\varepsilon^2 \frac{1}{1-\alpha}>0$, write $t_n:= \lceil (1-\varepsilon) \log_2n \rceil$, and $l_n= \lfloor  2^{\frac{\alpha t_n (1+\varepsilon)}{1-\alpha}} \rfloor$. Set $k_n: = \lfloor \frac{n}{l_n}\rfloor$.
Let
\[
E = \{x\in [0,1): R^\alpha_{t_n}(x) < t_n\}
\]
and
\[
F_j =\{x \in [0,1): R^\alpha_{jl_n+1, jl_n+t_n}(x) < t_n\},\ \ 1 \leq j \leq k_n-1.
\]
Then
\begin{align*}
\{x \in [0,1): R^\alpha_n(x) < t_n\} &\subseteq E \bigcap F_1 \bigcap \cdots \bigcap F_{k_n-1}\\
&= E \bigcap T^{-l_n}(E\cap F_1\cap \cdots \cap F_{k_n-2}).
\end{align*}

Note that $I_{t_n}(1,\cdots,1) \subseteq [1/2, 1)$ and $I_{t_n}(1,\cdots,1)=E^c$, where
\[
I_{t_n}(1,\cdots,1):=\{x \in [0,1):\varepsilon_1(x) = \cdots = \varepsilon_{t_n}(x)=1\}.
\]
By (\ref{mu0}), it follows that
\[
\widetilde{C}^{-1} \frac{1}{2^{t_n}}\leq \mu(I_{t_n}(1,\cdots,1)) \leq \widetilde{C}\frac{1}{2^{t_n}},
\]
where $\widetilde{C}>1$ is a constant.
Combing this with (\ref{mixing}), we deduce that
\begin{align*}
\mu\{x \in [0,1): R^\alpha_n(x) < t_n\} &\leq \mu(E \bigcap F_1 \bigcap \cdots \bigcap F_{k_n-1}) \notag\\
&\leq \left(c_{l_n}\mu(E)+ \frac{1}{l_n^{1/\alpha}} \right) \mu (E\cap F_1\cap \cdots \cap F_{k_n-2}) \notag\\
& \leq \cdots \cdots \notag\\
&\leq \left(c_{l_n}\mu(E)+ \frac{1}{l_n^{1/\alpha}} \right)^{k_n-1} \\
 &\leq \left(\left(1+ \frac{s}{2^{t_n(1+\varepsilon)}} + o(\frac{1}{2^{t_n(1+\varepsilon)}})\right)\left(1- \widetilde{C}^{-1} \frac{1}{2^{t_n}}\right)+ \frac{1}{2^{\frac{t_n(1+\varepsilon)}{1-\alpha}}}\right)^{k_n-1}\\
 & \leq \left(1- \frac{\widetilde{C}^{-1} }{2^{t_n}} + o(\frac{1}{2^{t_n}}) \right)^{k_n-1} \\
 & \leq \exp\left\{ -\frac{\widetilde{C}^{-1}}{ 2^{t_n}} \frac{n}{2^{\frac{t_n \alpha(1+\varepsilon)}{1 -\alpha}}}    \right\}.
\end{align*}
Then the choosing of $\varepsilon$ implies that

\[
\sum_{n \geq 1} \mu\{x \in [0,1): R^\alpha_n(x) < t_n\} <\infty,
\]
by Borel-Cantelli lemma, we conclude that
for $\mu$-almost all $x \in [0,1)$,
\[
\liminf_{n \to \infty} \frac{R^\alpha_n(x)}{\log_2 n} \geq 1-\varepsilon.
\]
Letting $\varepsilon \to 0^+$, we complete the proof.

\end{proof}

\section{Discussions for further research}
In this paper we have studied the run length function for the intermittent maps $T_{\alpha}$ with $0<\alpha<1$. A natural question is to estimate the run length function for intermittent maps with $\alpha\geq1$. Notice that an intermittent map $T_{\alpha}$ admits a unique ergodic invariant probability measure absolutely continuous with respect to Lebesgue measure if and only if $0<\alpha<1$, and in this case, the decay of correlation is polynomial. As we have mentioned in the Introduction, such polynomial decay of correlations and the classical Borel-Cantelli lemma are two fundamental ingredients in our proofs of the main results.

The difficulties in obtaining pointwise estimation of the run length function for intermittent maps with $\alpha \geq 1$ are twofold. Firstly, the decay of correlation of such maps do not hold. To be more precise, when $\alpha\geq 1$, each intermittent map admits a unique (up to scaling) $\sigma$-finite (but not finite), absolutely continuous invariant measure $\mu$. In these infinite measure settings, although one may still obtain certain mixing rate $\int_{[0,1]}fg\circ T^n_\alpha d\mu$ for some reasonably well-behaved observables $f,g$. (See e.g.,\cite{MT12, Ter16} for the concrete statements), there is no analogous result as in \eqref{equ:polynimal} to the best of our knowledge. Secondly, the classical Borel-Cantelli lemma does not hold in the infinite measure setting. Therefore, our methods in this paper are inapplicable to obtain the corresponding results for $\alpha\geq 1$ directly, and we expect to develop more new methods and techniques to overcome these difficulties.

\section*{Acknowledgments}
We would like to thank Prof. Manfred Denker and Prof. Huyi Hu for useful discussions, particularly for the proof of part I of our main theorem. This work is partially supported by grants from National Natural Science Foundation of China (Nos. 11701200, 11671395, 11801591, 11871262), Hubei Chenguang Talented Youth Development Foundation 2017 (Nos. 0106011025), and Ky and Yu-Fen Fan traveling award, National Science Foundation 2018.