TITLE: Create a method to prove that there are infinitely many numbers for a certain criteria
QUESTION [0 upvotes]: I have been facing some problems recently, when I need to prove that there are infinitely many natural numbers with some conditions.
It is very hard, because usually we can prove that of one or more conditions meet this criteria, however it will not neccesarily mean that the statement is true for all criteria at the same time. To compare different infinities is not easy.
I thought I would ask your help on a simple problem and learn how you would approach a problem like this.
I am interested in as many approaches as possible, be open minded,  because I would like to use these methods for more end more complicated problems and you never know what the best approach will be for those.
My example is simple: prove that there are infinitely many natural numbers that can not be written as either 2x, or 5x,  where x is a natural number.
Here we do have infinitely many natural numbers that are not 2x,  and we do have infinitely many natural numbers that are not 5x.
How would you prove the example?

REPLY [0 votes]: Take a natural number $a$ that is not a multiple of $2$ and is not a multiple of $5$. Then its square $a^2$ is also not a multiple of $2$ and is not a multiple of $5$ ($2$ and $5$ share a property that makes this true - think about what that property is).
Let $S$ be the set of natural numbers that are not a multiple of $2$ and not a multiple of $5$. If $S$ were finite than it would have a largest member $s_{max}$. But we have just shown above that $s_{max}^2$ would also be a member of $S$ and $s_{max}^2 > s_{max}$ (well, as long as $s_{max} >1$, but this is trivial). So this contradicts the assertion that $s_{max}$ is the largest member of $S$. Hence $S$ cannot be finite.
The general strategy here for proving a set $S$ of natural numbers with property $P$ is infinite is to assume $S$ is finite, and then find or construct a number that has property $P$ but is not in $S$, typically because it is greater than  the largest member of $S$. This proves that $S$ cannot be finite.


text:
\begin{document}
\title{A sequence of radially symmetric weak solutions for some nonlocal elliptic problem in $\mathbb{R}^N$}
\author[a]{M. Makvand Chaharlang\thanks{moloudmakvand@gmail.com}}
\author[b]{M. A. Ragusa\thanks{maragusa@dmi.unict.it}}
\author[a]{A. Razani\thanks{razani@sci.ikiu.ac.ir}}
\affil[a]{Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Postal code: 34149-16818, Qazvin, Iran.}
\affil[b]{Department of Mathematics, University of Catania, Viale
Andrea Doria No. 6, Catania 95128, Italy, and RUDN University, Miklukho-Maklaya Str. 6, Moscow, 117198, Russia.}
\date{\empty}
\maketitle
\begin{abstract}
In this article a nonlocal elliptic problem involving $p$-Laplacian on unbounded domain is considered.
Using variational methods and under suitable conditions, the existence of a sequence of radially symmetric weak solutions (in two different cases) is proved.
\begin{description}
\item[Keywords:] Sequence of solutions, nonlocal elliptic problem, $p$-Laplacian, variational methods.
\end{description}
$2010$ Mathematics subject classification: 35J20, 34B10, 35J50.
\end{abstract}
\section{Introduction}
The theory of partial differential equations is a one of the powerful theories in mathematics.  On one side, this theory is used
 to model a wide variety
of physically significant problems arising in every different areas such as physics, engineering and other
applied disciplines (see \cite{Ehsani,hesaraki01,hesaraki02,Lindgren,Mokhtar,Mokhtarp,pournaki,pournakir,RaggusaDMJ, razani021, razani04,razani041,razani,razani19,razani02,razanigo,ScapllatoBVP}), and on the other side PDE's can be considered as an
instrument in the development of other branches of mathematics.

Many problems of mathematical physics and variational calculus are often very complicated. In all of them it is not
sufficient to deal with the classical solutions of differential equations. In fact, in the twentieth century, however, it was observed
that the space $C^k$ was not exactly the right space to
study solutions of differential equations. Although, solutions in $C^k$ are nicer but it seems people are happy with obtaining weak
solutions in some Sobolev space which are the modern replacement for these spaces that only satisfy the weak formulation. Therefore,
Sobolev spaces play an important role in the theory of partial differential equations. Since then, the study of the weak solution in other spaces such as
Orlicz-Morrey space and  $\dot{B}_{\infty,\infty}^{-1}$ space is a research problem (see \cite{GalaApplicable, GalaCMA, GalaAIMS}).

Laplace equation is the prototype for linear elliptic equations, as the most important partial differential equation of the second order.
This equation has a non-linear counterpart, the so-called $p$-Laplace
equation (see \cite{BehboudiFilomat, KhalkhaliInfinitely, Khalkhali2013, MahdaviFilomat,MakvandTJM,MakvandFilomat,MakvandGMJ}).
There has been a surge of interest in the $p$-Laplacian in many
different contexts from game theory to mechanics and image processing. Recently, a great attention has been focused on the study of
nonlocal operators of elliptic type (for example \cite{MakvandTJM,MakvandCKMS}), both, for research in pure Mathematics and for concrete
real world applications. From a physical point of view, the nonlocal operators play a crucial role in describing
several phenomena such as, the thin obstacle problem, optimization, material science,  geophysical fluid dynamics
and mathematical finance, phase transitions, water waves.

In this article we consider the following problem
\begin{equation}\label{eq1.1}
\begin{cases}
-M\left(\int_\mathbb{R^N} |\nabla u(x)|^p\,dx + \int_\mathbb{R^N}|u(x)|^pdx\right)(\Delta_p u+|u|^{p-2}u)=\lambda f(x,u) & \text{in $\mathbb{R}^N$},\\
 u\in W^{1,p}(\mathbb{R}^N)
\end{cases}
\end{equation}
where $p> N > 1$, $(\mathbb{R}^N,|.|)$ is the usual Euclidean space and $M:[0,+\infty[ \rightarrow \mathbb{R} $ is a continuous function and there exist two positive
constants $m_0$ and $m_1$ such that $m_0 \leq M(t) \leq m_1$ for all $t \geq 0$. We assume that
$f:\mathbb{R}^N \times \mathbb{R}\to \mathbb{R}$ is an $L^1$-Carath\'eodory function and $\lambda >0 $ is a real parameter.
Also $\Delta_pu:=\mr{div}(|\nabla u|^{p-2}\nabla u)$ denotes the $p$-Laplacian operator.
This equation is related to the stationary version of the Kirchhoff equation
\begin{equation}\label{eq23}
\rho \dfrac{\partial ^2 u}{\partial t^2} - (\dfrac{P_0}{h}+\dfrac{E}{2L}\int_0^L |
\dfrac{\partial u}{\partial x}|^2 \mr{d}x )\dfrac{\partial ^2 u}{\partial x^2}=0,
\end{equation}
proposed by Kirchhoff in 1883. This equation is an extension of the classical
d'Alembert's wave equation by considering the effects of the length changes of the string produced by
transverse vibrations. Since the first equation in \eqref{eq1.1} contains an integral over $\mathbb{R}^N $,
it is no longer a pointwise identity, and therefore it is often called a nonlocal problem. This problem models
several physical and biological systems, where $u$ describes a process which depends on the average of itself,
such as the population identity, (for example see \cite{chipot}). The parameters in \eqref{eq23} have the following meanings:
\emph{$h$} is the cross-section area, \emph{$E$} is the Young modulus, \emph{$\rho $} is the mass density, \emph{$L$}
is the length of the string, and \emph{$P_0$} is the initial tension. In recent years, $p$-Kirchhoff type problems have
been studied by many researchers, (for example see \cite{MakvandTJM,MakvandCKMS}). Beside elliptic problems with boundary conditions on
bounded domain of $\mathbb{R}^N$ which have extensive applications in different parts of scient and have been considered
by many authors recently, see \cite{MakvandGMJ,MakvandFilomat}, some elliptic problems arise on unbounded domain $\mathbb{R}^N$, see
\cite{Chen,Yang}. It is worth mentioning that one of the difficulties in studying problems on
unbounded domains is that there is no compact embedding for $W^{1,p}(\mathbb{R}^N)$. Although, we can use the continuous embedding
$W^{1,p}(\mathbb{R^N})\hookrightarrow L^{\infty}(\mathbb{R^N})$ from Morrey's theorem, it is far from being compact. Considering,
 a crucial embedding result due to Krist\'{a}ly and principally
based on a Strauss-type estimation which shows $W_r^{1,p}(\mathbb{R}^N)\hookrightarrow L^{\infty}(\mathbb{R}^N)$ is compact
whenever $2\leq N<p<+\infty$, where $W_r^{1,p}(\mathbb{R}^N)$ is a subspace of radially symmetric functions of $W^{1,p}(\mathbb{R}^N)$,
we prove the existence of a sequence of radially symmetric weak solutions for \eqref{eq1.1} which is a
nonlocal elliptic problem in the unbounded domain $\mathbb{R}^N$.

\section{Preliminaries}
In this part we remind some definitions and theorems as follows:
\begin{definition}\label{defn1}
The function $f:\mathbb{R}^N \times \mathbb{R}\to \mathbb{R}$ is said to be a\\
$L^1$-Carath\'eodory function, if
\begin{description}
\item[($A_1$)] the function $x\mapsto f(x,t)$ is measurable for every $t\in \mathbb{R}$,
\item[($A_2$)] the function $t\mapsto f(x,t)$ is continuous for a.e. $x\in \mathbb{R}^N$,
\item[($A_3$)] for every $h>0$ there exists a function $\ell_h\in L^1(\mathbb{R}^N)$ such that\\ $\sup_{|t|\leq h}|f(x,t)|\leq \ell_h(x)$,
for a.e. $x\in \mathbb{R}^N$.
\end{description}
\end{definition}

\begin{definition}
For fixed $\lambda$, a function $u:{\mathbb{R}^N}\rightarrow\mathbb{R}$ is said to be a weak solution of \eqref{eq1.1},
if $u \in W^{1,p}\left(\mathbb{R}^N\right)$ and for every $v\in W^{1,p}\left(\mathbb{R}^N\right)$
\[
\begin{array}{r}
M(\|u\|_{W^{1,p}\left(\mathbb{R}^N\right)})(\int_{\mathbb{R}^N}|\nabla
u(x)|^{p-2}\nabla u(x).\nabla v(x)dx +\int_{\mathbb{R}^N}|u(x)|^{p-2}u(x)v(x)dx)\\
-\lambda\int_{\mathbb{R}^N}f(x,u(x))v(x)dx=0,
\end{array}
\]
where $\|u\|_{W^{1,p}\left(\mathbb{R}^N\right)}:=\left(\int_{\mathbb{R}^N} |\nabla u(x)|^p dx + \int_\mathbb{R^N}|u(x)|^pdx\right)^{1/p}$.
\end{definition}

Later on, w define a functional energy where its the critical points are exactly the weak solutions of \eqref{eq1.1}.

Morrey's theorem, implies the continuous embedding
\begin{equation}\label{embedding}
W^{1,p}(\mathbb{R^N})\hookrightarrow L^{\infty}(\mathbb{R^N}),
\end{equation}
which says that there exists $c:=\frac{2p}{p-N}$, such that
$\|u\|_{\infty}\leq c\|u\|_{W^{1,p}(\mathbb{R}^N)}$, for every $u \in W^{1,p}(\mathbb{R^N})$, where $\|u\|_\infty:=\underset{x\in \mathbb{R}^N}{esssup}|u(x)|$, for every $u\in L^\infty(\mathbb{R}^N)$. Since in the low-dimensional case, every function $u\in W^{1,p}(\mathbb{R}^N)$
admits a continuous representation (see \cite[p.166]{Brezis}. In the sequel we will replace $u$ by this element.

We need the following notations (see \cite{candito} for more details):
\begin{itemize}
\item[(I)] $O(N)$ stands for the orthogonal group of $\mathbb{R}^N$.
\item[(II)] $B(0,s)$ denotes the open $N$-dimensional ball of center zero, radius $s>0$ and standard Lebesgue measure, $meas(B(0,s))$.

\end{itemize}

\begin{definition}\
\begin{itemize}
\item A function $h : \mathbb{R}^N\to \mathbb{R}$ is radially symmetric if $h(gx) = h(x)$, for every
$g\in O(N)$ and $x\in \mathbb{R}^N$.
\item Let $G$ be a topological group. A continuous map $\xi:G\times X\to X:(g,x)\to \xi(g,u):=gu$,
is called the action of $G$ on the Banach space $(X,\|.\|_X)$ if
\[
1u=u,\qquad (gm)u=g(mu),\qquad u\mapsto gu\ \text{is linear}.
\]
\item  The action is said to be isometric if $\|gu\|_X=\|u\|_X$, for every $g\in G$.
\item  The space of $G$-invariant points is defined by
\[
Fix(G):=\{ u\in X: gu=u, \text{for all}\ {g\in G}\}.
\]
\item A map $m:X\to \mathbb{R}$ is said to be $G$-invariant if $mog=m$ for every $g\in G$.
\end{itemize}
\end{definition}
The following theorem is important to study the critical point of the functional.
\begin{theorem}\label{palais}
(Palais (1979)) Assume that the action of the topological group $G$ on the Banach space $X$ is isometric.
If $J\in C^1(X:\mathbb{R})$ is $G$-invariant
and if $u$ is a critical point of $J$ restricted to $Fix(G)$, then $u$ is a critical point of $J$.
\end{theorem}

The action of the group $O(N)$ on $W^{1,p}(\mathbb{R}^N)$ can be defined by $(gu)(x):=u(g^{-1}x)$, for every
$g\in W^{1,p}(\mathbb{R}^N)$ and $x\in \mathbb{R^N}$.
A computation shows that this group acts linearly and isometrically, which means $\|u\|=\|gu\|$, for every
$g\in O(N)$ and $u\in W^{1,p}(\mathbb{R}^N)$.

\begin{definition}
The subspace of radially symmetric functions of $W^{1,p}(\mathbb{R^N})$ is defined by
\[
X:=W_r^{1,p}(\mathbb{R}^N):=\{ u\in W^{1,p}(\mathbb{R}^N): gu=u,\ \text{for all}\ g\in O(N)\},
\]
and endowed by the norm $\|u\|_r:=\left(\int_\mathbb{R^N} |\nabla u(x)|^p dx + \int_\mathbb{R^N}|u(x)|^pdx\right)^{1/p}$.
\end{definition}
The following crucial embedding result due to Krist\'{a}ly and principally based on a Strauss-type estimation (see \cite{Strauss})
(Also see \cite[Theorem 3.1]{Kristaly}, \cite{Varga} and \cite{Willem} for related subjects).
\begin{theorem}\label{thcom}
The embedding $W^{1,p}_r(\mathbb{R}^N)\hookrightarrow L^{\infty}(\mathbb{R}^N)$, is compact whenever $2\leq N<p<+\infty$.
\end{theorem}

Here we consider the following functionals:
\begin{itemize}
\item $F(x,\xi):=\int_{0}^{\xi}f(x,t)\mr{d}t,\quad $ for every
$(x,\xi)\in\mathbb{R^N}\times\mathbb{R}$,
\item $\widehat{M}(t):= \int_0^t M(s) \mr{d}s ,\quad$ for every $t>0$,
\item $\Phi(u):=\frac{1}{p}\widehat{M}(\|u\|_r^p) $ for every $u \in X$,
\item $\Psi(u):=\int_\mathbb{R^N}F(x,u(x))\mr{d}x$, for every $u \in X$,
\item $I_\lambda(u):=\Phi(u)-\lambda \Psi(u)$ for every $u \in X$.
\end{itemize}
By standard arguments \cite{candito}, we can show that $\Phi$ is G\^ateaux differentiable, coercive and
sequentially weakly lower semicontinuous whose derivative at the point $u \in X$ is the functional $\Phi'(u)\in X^*$ given by
\[
\Phi'(u)(v)=M(\|u\|^p_r)(\int_\mathbb{R^N}|\nabla u|^{p-2}\nabla u.\nabla v dx + \int_\mathbb{R^N}|u|^{p-2}uv dx),
\]
for every $v\in X$. Also standard arguments show that the functional $\Psi$ is well defined,
sequentially weakly upper semicontinuous and G\^ateaux differentiable whose G\^ateaux derivative
at the point $u \in X$ and for every $v\in X$ is given by,
\[
\Psi'(u)(v)=\int_\mathbb{R^N}f(x,u(x)) dx.
\]
In the last part of this section we recall the
following theorem \cite[Theorem 2.1]{bonanno}.
\begin{theorem}\label{theo2.1}
Let $X$ be a reflexive real Banach space, let $\Phi,\Psi:X\to \mathbb{R}$ be two G\^ateaux differentiable
functionals such that $\Phi$ is sequentially weakly lower semicontinuous, strongly continuous and coercive,
and $\Psi$ is sequentially weakly upper semicontinuous. For every $r>\inf_X\Phi$, put
\[
\begin{aligned}
\varphi(r)&:=\inf_{\Phi(u)<r}\frac{\sup_{\Phi(v)<r}\Psi(v)-\Psi(u)}{r-\Phi(u)},\\
\gamma&:=\liminf_{r\to+\infty}\varphi(r),\quad \text{and}\quad \delta:=\liminf_{r\to(\inf_X\Phi)^+}\varphi(r).
\end{aligned}
\]
Then the following properties hold:
\begin{itemize}
\item[($a$)] for every $r>\inf_X\Phi$ and every $\lambda \in]0,\frac{1}{\varphi(r)}[$, the restriction of the functional
\[ I_\lambda:=\Phi-\lambda \Psi \]
to $\Phi^{-1}(]-\infty,r[)$ admits a global minimum, which is a critical point (local minimum) of $I_\lambda$ in $X$.
\item[($b$)] if $\gamma<+\infty$, then for each $\lambda \in]0,\frac{1}{\gamma}[$, the following alternative holds either,
\begin{itemize}
\item[($b_1$)] $I_\lambda$ possesses a global minimum, or
\item[($b_2$)] there is a sequence $\{u_n\}$ of critical points (local minima) of $I_\lambda$ such that
\[ \lim_{n\to+\infty}\Phi(u_n)=+\infty, \]
\end{itemize}
\item[($c$)] if $\delta<+\infty$, then for each $\lambda \in]0,\frac{1}{\delta}[$, the following alternative holds either:
\begin{itemize}
\item[($c_1$)] there is a global minimum of $\Phi$ which is a local minimum of $I_\lambda$. or,
\item[($c_2$)] there is a sequence $\{u_n\}$ of pairwise distinct critical points (local minima) of
$I_\lambda$ which weakly converges to a global minimum of $\Phi$, with
\[ \lim_{n\to+\infty}\Phi(u_n)=\inf_{u\in X}\Phi(u). \]
\end{itemize}
\end{itemize}
\end{theorem}
\section{Multiple solutions}
In this part in order to present our main results we consider some assumptions as follows.
For fixed $D>0$, let
\[m(D):=meas(B(0,D))=D^N\frac{\pi^{\frac{N}{2}}}{\Gamma(1+\frac{N}{2})},\]
 where
$\Gamma$ is the Gamma function defined by $\Gamma(t):=\int_0^{+\infty}z^{t-1}e^{-z}dz$ for all $t>0$.  Moreover,
\begin{equation}\label{k}
 k:=\dfrac{1}{m_1 m(D){c^p}\left(\frac{\sigma(N,p)}{D^p}+g(p,N)\right)}\geq 0,
\end{equation}
where $\sigma(N,p):=2^{p-N}(2^N-1)$, $c= \frac{2p}{2-N}$, $m_1,m_0$ are upper and lower bounds for $M(t)$ in \eqref{eq1.1} and
\[g(p,N):=\frac{1+2^{N+p}N\int_{\frac{1}{2}}^1
t^{N-1}(1-t)^pdt}{2^N}.\]
Let $\|\cdot\|_1$ denotes the norm of $L^1(\Omega)$, we present our first result as the following theorem.
\begin{theorem}\label{theo1}
Let $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ be an
$L^1$- Carath\'{e}odory function, $f$ is radially symmetric with respect to first component, $F(x,t)\geq0$ for every
$(x,t)\in\mathbb{R^N}\times\mathbb{R^+}$ and $A<km_0 B$, where
$A:=\liminf_{\xi\rightarrow \infty}\frac{\|\ell_\xi\|_1}{\xi^{p-1}}$, $B:=\limsup_{\xi\rightarrow \infty}
\frac{\int_{B(0,\frac{D}{2})}F(x,\xi)dx}{\xi^p}$, $\ell_\xi\in L^1(\mathbb{R^N})$  and $k$ are given
by \eqref{defn1} and \eqref{k}, respectively. Then
for every
$$\lambda\in \Lambda:=\left]\frac{m_1 m(D)}{B}\left(\frac{\sigma(N,p)}{pD^p}+\frac{g(p,N)}{p}\right),\frac{m_0}{p c^p A}\right[,$$
there exists an unbounded sequence of radially symmetric weak solutions for \eqref{eq1.1} in $X$.
\end{theorem}
\begin{proof}
For fixed $\lambda \in \Lambda$, we consider $\Phi$, $\Psi$ and $I_\lambda$ as in
Section 2. Knowing that $\Phi$ and
$\Psi$ satisfy the regularity assumptions in Theorem \eqref{theo2.1}.  In order to study
the critical points of $I_\lambda$ in $X$, we show that $\lambda <\frac{1}{\gamma}<+\infty$, where $\gamma=\underset{r\to +\infty}{\liminf}\phi(r)$.

Let $\lbrace t_n \rbrace $ be a sequence of positive numbers such that $\lim_{n\to \infty} t_n=+\infty $ and
\[A=\lim_{n\rightarrow +\infty}\frac{\|\ell_{t_n}\|_1}{{t_n}^{p-1}}.\]
Set $r_n:=\dfrac{m_0{t_n}^p}{pc^p}$, for all $n \in \mr{N}$. Considering Theorem \ref{thcom} (by relation \eqref{embedding}), one has
\begin{equation}\label{phii}
\begin{aligned}
\Phi^{-1}(]-\infty,r_n[)&=\{v\in X;\, \Phi(v)<r_n\}\\
& = \{v\in X;\ \|v\|_r<(\frac{pr_n}{m_0})^{\frac{1}{p}}\}\\
& \subset \{v\in X;\ \|v\|_{\infty}<t_n\}.
\end{aligned}
\end{equation}
Since $\Phi(0)=\Psi(0)=0$, we have
\[
\begin{aligned}
\varphi(r_n)&=\inf_{\Phi(u)<r_n}\frac{(\sup_{\Phi(v)<r_n}\Psi(v))-\Psi(u)}{r_n-\Phi(u)}\\
& \leq \frac{\sup_{\|v\|_{\infty}<t_n}\int_{\mathbb{R}^N} F(x,v(x))dx}{\dfrac{m_0{t_n}^p}{pc^p}}\\
& \leq \frac{pc^p}{m_0}\frac{\|\ell_{t_n}\|_1}{{t_n}^{p-1}}.
\end{aligned}
\]
Hence, it follows that
\[
\gamma= \liminf_{n\to +\infty}\varphi(r_n)\leq \frac{pc^p}{m_0}\liminf_{n\to+\infty}\frac{\|\ell_{t_n}\|_1}{t_n^{p-1}}=\frac{pc^p}{m_0}A<+\infty.
\]
Also, from $\lambda \in \Lambda$, one has
$\dfrac{1}{\lambda} >\frac{pc^p}{m_0}A,$ and we get $\lambda <\dfrac{1}{\gamma}$.\\
Now, we show that $I_\lambda$ is unbounded from below. Let $\lbrace d_n\rbrace$ be a sequence of positive numbers such that $\lim_{n \rightarrow +\infty}d_n=+\infty$ and
\begin{equation}\label{B}
B=\lim_{n\rightarrow +\infty}
\frac{\int_{B(0,\frac{D}{2})}F(x,d_n)dx}{{d_n}^p}.
\end{equation}
Assume $\{v_n\} \subset X$ defined by
\begin{equation*}
 v_n(x):=\left\{\begin{array}{lll}
\displaystyle0  & \quad \mathbb{R^N}\setminus B(0,D)\\
\displaystyle d_n  &  \quad B(0,\frac{D}{2})
\\
\displaystyle \frac{2d_n}{D}(D-|x|)  & \quad B(0,D)\setminus
B(0,\frac{D}{2}),
\end{array}\right.
\end{equation*}
for every $n \in \mr{N}$. By a similar argument and compuations in \cite[P.1017]{candito} one can show that
$$\|v_n\|_r^p=d_n^pm(D)\left(\frac{\sigma(N,p)}{D^P}+g(p,N)\right).$$
Condition $(i)$, implies
$$\int_{\mathbb{R^N}}F(x,v_n(x))dx\geq\int_{B(0,\frac{D}{2})}F(x,d_n)dx,$$
for every $n \in N$. Therefore
\[\Phi(v_n)\leq \frac{m_1}{p}\|v_n \|_r^p=\frac{m_1 }{p}m(D) d_n^p\left(\frac{\sigma(N,p)}{D^p}+g(p,N)\right).\]
Let
\[\alpha:=m_1m(D)\left(\frac{\sigma(N,p)}{pD^P}+\dfrac{g(p,N)}{p}\right). \]
If $B <+\infty $,  for every $\varepsilon \in (\dfrac{\alpha}{\lambda B},1)$  and
$\varepsilon':=(1-\varepsilon)B >0$, \eqref{B} implies that there exists $N_\varepsilon$ such that for all $n>N_\varepsilon$,
\[
(\varepsilon-1)B <\frac{1}{d_n^p}\int_{B(0,\frac{D}{2})}F(x,d_n)dx - B<(1-\varepsilon) B.
\]
So
\[
\int_{B(0,\frac{D}{2})}F(x,d_n)dx>\varepsilon Bd_n^p,\qquad \text{for all}\ n>N_\varepsilon.
\]
Therefore for every $n> N_\varepsilon$,
\[
\begin{aligned}
I_\lambda(v_n)&\leq m_1d_n^pm(D)\left(\frac{\sigma(N,p)}{pD^p}+\dfrac{g(p,N)}{p}\right)-\lambda\varepsilon Bd_n^p\\
&=d_n^p(\alpha -\lambda \varepsilon B).
\end{aligned}
\]
Hence, $\lim_{n\rightarrow\infty}I_\lambda(v_n)=-\infty.$
\\If $B=+\infty$, let $L>\dfrac{\alpha}{\lambda}$. From \eqref{B} there exists $N_L$ such that for all $n>N_L$,
\[\int_{B(0,\frac{D}{2})}F(x,d_n)dx>Ld_n^p.\]
Thus
\[I_\lambda (v_n)\leq d_n^p\alpha-\lambda Ld_n^p=d_n^p(\alpha-\lambda L), \]
for every $n>N_L$, which shows that
\[\lim_{n\rightarrow\infty}I_\lambda(v_n)=-\infty.\]
Now, Theorem \ref{theo2.1} ($b$) implies, the functional $I_\lambda$ admits an unbounded sequence $\{u_n\}\subset X$ of
 critical points. Considering Theorem \ref{palais}, these critical points are also critical points for the smooth and $O(N)$-invariant functional $I_\lambda:W^{1,p}\left(\mathbb{R}^N\right)\to \mathbb{R}$. Therefore, there is a sequence of radially semmetric weak soluitions for the problem \eqref{eq1.1}, which are unbounded in $W^{1,p}\left(\mathbb{R}^N\right)$.
\end{proof}
Here we prove our second result which says that under different conditions the problem \eqref{eq1.1} has a sequence of
weak solutions, which converges weakly to zero.
\begin{theorem}\label{3.1}
Let $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ be an
$L^1$- Carath\'{e}odory function, $f$ is radially symmetric with respect to first component, $F(x,t)\geq0$ for every
$(x,t)\in\mathbb{R^N}\times\mathbb{R^+}$ and $A'<km_0 B'$
where
$ A':=\liminf_{\xi\rightarrow 0^+}\frac{\|\ell_\xi\|_1}{\xi^{p-1}}$, $B':=\limsup_{\xi\rightarrow 0^+}
\frac{\int_{B(0,\frac{D}{2})}F(x,\xi)dx}{\xi^p}$,
$\ell_\xi\in L^1(\mathbb{R^N})$  and $k$ are given by \eqref{defn1} and  \eqref{k}, respectively. Then
for every
$$\lambda\in \Lambda':=\left]\frac{m_1m(D)}{B'}\left(\frac{\sigma(N,p)}{pD^p}+\frac{g(p,N)}{p}\right),\frac{m_0}{p c^p A'}\right[,$$
there exists a sequence of weak solutions for \eqref{eq1.1} which converges weakly to zero in $X$.
\end{theorem}
\begin{proof}
For fixed $\lambda \in \Lambda'$, we consider $\Phi$, $\Psi$ and $I_\lambda$ as in Section 2. Knowing that
$\Phi$ and $\Psi$ satisfy the regularity assumptions in Theorem \eqref{theo2.1}, we show that $\lambda <\frac{1}{\delta}$. We know that $\inf_X\Phi =0$, therefore,
\[\delta:=\liminf_{r\to 0^+}\varphi(r).\]
Let $\lbrace t_n \rbrace $ be a sequence of positive numbers such that $\lim_{n\to \infty} t_n=0 $ and
\[A'=\lim_{n\rightarrow +\infty}\frac{\|\ell_{t_n}\|_1}{{t_n}^{p-1}}\]
Set $r_n:=\dfrac{m_0{t_n}^p}{pc^p}$, for all $n \in \mr{N}$. Considering \eqref{phii}, one has
$\|v\|_{\infty}<t_n$ for every $v\in X$ with $\Phi(v)<r_n$.
Since $\Phi(0)=\Psi(0)=0$, we have
\[
\begin{aligned}
\varphi(r_n)&=\inf_{\Phi(u)<r_n}\frac{(\sup_{\Phi(v)<r_n}\Psi(v))-\Psi(u)}{r_n-\Phi(u)}\\
&\leq \frac{\sup_{\|v\|_{\infty}<t_n}\int_{\mathbb{R}^N} F(x,v(x))dx}{\dfrac{m_0{t_n}^p}{pc^p}}\\
&\leq \frac{pc^p}{m_0}\frac{\|\ell_{t_n}\|_1}{{t_n}^{p-1}}.
\end{aligned}
.\]
Hence, it follows that
\[\delta \leq \liminf_{n\to +\infty}\varphi(r_n)\leq pc^p\liminf_{n\to+\infty}\frac{\|\ell_{t_n}\|_1}{t_n^{p-1}}=pc^pA'<+\infty.\]
Also, from $\lambda \in \Lambda'$, one has
$\dfrac{1}{\lambda} > pc^pA',$ and we get $\lambda <\dfrac{1}{\delta}$.\\
Now, we show that zero is not a local minimum of $I_\lambda. $ Let $\lbrace d_n\rbrace$ be a sequence of
positive numbers such that $\lim_{n \rightarrow +\infty}d_n=0$ and
\begin{equation}\label{BB}
B=\lim_{n\rightarrow +\infty}
\frac{\int_{B(0,\frac{D}{2})}F(x,d_n)dx}{{d_n}^p}.
\end{equation}
Also consider $\{v_n\}\subset X$ similar to the proof of Theorem \ref{theo1}, defined by $\{d_n\}$ above. By the
same argument, we obtain that $I_\lambda(v_n)<0$ for $n$ large enough. Thus zero is not a local minimum of $I_\lambda$ as
 $\lim_{n\to+\infty}I_\lambda(v_n)<I_\lambda(0)=0$. Therefore, there exists a sequence $\{u_n\}\subset X$ of
critical points of $I_\lambda$ which converges weakly to zero in $X$ as  $\lim_{n\to+\infty}\Phi(u_n)=0$.

Again, Considering Theorem \ref{palais}, these critical points are also critical points for the smooth and $O(N)$-invariant functional $I_\lambda:W^{1,p}\left(\mathbb{R}^N\right)\to \mathbb{R}$. Therefore, there is a sequence of radially symmetric weak solutions for the problem \eqref{eq1.1}, which converges weakly to zero in $W^{1,p}\left(\mathbb{R}^N\right)$.
\end{proof}

\section*{Acknowledgements}
We would like to thank the anonymous reviewer for the careful reading of our manuscript and the valuable comment and suggestion.\\
The second author is partially supported by
I.N.D.A.M - G.N.A.M.P.A. 2019 and the ``RUDN University Program 5-100''.\\
This work is done when the third author is visiting University of Manitoba on a sabbatical leave from Imam Khomeini International University.

config: arxiv


text:
\begin{document}

\title{Banach spaces and Ramsey Theory: some open problems}
\author{Pandelis Dodos, Jordi Lopez-Abad and Stevo Todorcevic}

\address{Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece.}
\email{pdodos@math.uoa.gr}

\address{Instituto de Ciencias Matematicas (ICMAT), CSIC-UAM-UC3M-UCM, Madrid, Spain.}
\email{abad@logique.jussieu.fr}

\address{Universit\'{e} Denis Diderot - Paris 7, C.N.R.S., UMR 7056, 2 place Jussieu
- Case 7012, 72521 Paris Cedex 05, France}
\address{and}
\address{Department of Mathematics, University of Toronto, Toronto, Canada, M5S 2E4.}
\email{stevo@math.toronto.edu}

\thanks{2010 \textit{Mathematics Subject Classification}: 03E02, 03E55, 05D10,
46B03, 46B07, 46B15, 46B26, 46B80, 47B38}
\thanks{\textit{Key words}: Banach spaces, Ramsey Theory, open problems}

\maketitle

\begin{abstract}
We discuss some open problems in the Geometry of Banach spaces having Ramsey-theoretic
flavor. The problems are exposed together with well known results related to them.
\end{abstract}

\section{Introduction}

One of the most illuminating imports in Banach Space Theory is the field
of Ramsey Theory. There have been some remarkable achievements using combinatorial
techniques; see, for instance, \cite{AGR,AT,G3,Od} and the references therein.
The purpose of this paper is to collect a number of \textit{open problems}
in the Geometry of Banach spaces having Ramsey flavor. We have not tried
to be encyclopedic in our selection. Instead, we have chosen to discuss the
following list of open problems which shows, in particular, that Ramsey Theory
cuts across all parts of modern Banach Space Theory.
\begin{enumerate}
\item[$\bullet$] \textit{Distance from the cube}.
\item[$\bullet$] \textit{Elton unconditionality constant}.
\item[$\bullet$] \textit{Rosenthal basis problem}.
\item[$\bullet$] \textit{Unique spreading model problem}.
\item[$\bullet$] \textit{Block homogeneous basis problem}.
\item[$\bullet$] \textit{Fixing properties of operators on $C[0,1]$}.
\item[$\bullet$] \textit{Bounded distortion}.
\item[$\bullet$] \textit{Converse Aharoni problem}.
\item[$\bullet$] \textit{Separable quotient problem}.
\item[$\bullet$] \textit{Quotients with long Schauder bases and biorthogonal systems}.
\item[$\bullet$] \textit{Rolewicz problem on support sets}.
\item[$\bullet$] \textit{Unconditional basic sequences in non-separable spaces}.
\end{enumerate}
In several cases, a problem naturally leads to a number of related questions
which are interesting on their own. We discuss these issues in detail.

Our Banach space theoretic notation and terminology is standard and follows
\cite{LT}. By $\nn=\{0,1,2,...\}$ we shall denote the natural numbers. If
$S$ is a set, then the cardinality of $S$ will be denoted by $|S|$. Finally,
by $[\nn]^{\infty}$ we denote the set of all infinite subsets of $\nn$.

\section{Distance from the cube}

Recall that the \textit{Banach-Mazur distance} between two isomorphic Banach spaces
$X$ and $Y$ (not necessarily infinite-dimensional) is defined by
\[ d_{BM}(X,Y)=\inf\big\{\|T\|\cdot\|T^{-1}\|: T:X\to Y \text{ is an isomorphism}\big\}.\]
For every $n\in\nn$ define
\[ R^n_\infty=\max\big\{ d_{BM}(X,\ell^n_\infty): \mathrm{dim}(X)=n\big\}.\]
\begin{pro} \label{p1}
\emph{(Distance from the cube -- A. Pe{\l}czy\'{n}ski \cite{Pe2})}
Determine the asymptotic behavior of $R^n_\infty$ as $n$ tends to infinity.
\end{pro}
The best known lower bound for $R^n_\infty$ is due to S. Szarek \cite{Szarek}:
\[ c\sqrt{n}\log n\leqslant R^n_\infty \]
where $c$ is an absolute constant. The problem of finding good upper
estimates for $R^n_\infty$ turned out to be a very challenging problem and
has attracted the attention of several researchers; see \cite{BS,ST}.
The best known upper bound is due to A. Giannopoulos \cite{Gia}:
\[ R^n_\infty \leqslant C n^{5/6}\]
where $C$ is an absolute constant.

This problem appears in our list because all proofs establishing a non-trivial upper bound for $R^n_\infty$
use some variant of the \textit{Sauer-Shelah Lemma}. This basic combinatorial fact asserts that if $S$ is a
non-empty finite set and $\mathcal{F}$ is a ``large" family of subsets of $S$, then there exists a ``large"
subset $A$ of $S$ such that the \textit{trace $\mathcal{F}[A]$ of $\mathcal{F}$ on $A$}, i.e. the family
$\mathcal{F}[A]=\{F\cap A: F\in\mathcal{F}\}$, is the powerset of $A$.
\begin{thm} \label{t1}
\emph{(Sauer-Shelah Lemma \cite{Sauer,Sh1})} Let $S$ be a non-empty finite set, $k\in\nn$ with $1\leqslant
k\leqslant |S|$ and $\mathcal{F}$ be a family of subsets of $S$ such that
\[ |\mathcal{F}| > \sum_{i=0}^{k-1} \binom{|S|}{i}.\]
Then there exists a subset $A$ of $S$ with $|A|=k$ such that the trace $\mathcal{F}[A]$ of $\mathcal{F}$ on
$A$ is the powerset of $A$.
\end{thm}
The Sauer-Shelah Lemma turned out to be a very important tool in the ``local theory" of Banach spaces and has
found significant applications. It was first used in Banach Space Theory by J. Elton \cite{Elt2}.

\section{Elton unconditionality constant}

The Sauer-Shelah Lemma has an infinite version. To state it let us recall that a family $\mathcal{F}$ of finite
subsets of $\nn$ is said to be \textit{compact} (respectively, \textit{hereditary}) if $\mathcal{F}$ is a closed
subset of $2^\nn$ (respectively, for every $F\in\mathcal{F}$ and every $G\subseteq F$ we have that $G\in\mathcal{F}$).
\begin{thm} \label{t2}
\emph{(J. Elton -- see \cite{AGR})}
Let $\mathcal{F}$ be a compact family of finite subsets of $\nn$.
Then there exists $L\in[\nn]^\infty$ such that
$\mathcal{F}[L]$ is a hereditary family.
\end{thm}
This result can be seen as the discrete analogue of a heavily investigated
phenomenon in Banach Space Theory known as \textit{partial unconditionality}.

Recall that a normalized sequence $(x_n)$ is a Banach space $X$ is
said to be \textit{unconditional} if there exists a constant $C\geqslant 1$
such that for every $k\in\nn$, every $a_0,..., a_k\in \rr$ and
every $G\subseteq \{0,...,k\}$ we have
\begin{equation} \label{e1}
\big\| \sum_{n\in G} a_n x_n \big\| \leqslant C \big\| \sum_{n=0}^k a_n x_n \big\|.
\end{equation}
A classical discovery due to B. Maurey and H. P. Rosenthal \cite{MR}
provides an example of a normalized weakly null sequence $(e_n)$ in
a Banach space $E$ with no unconditional subsequence. A surprising
fact is, however, that for \textit{every} normalized weakly
null sequence $(x_n)$ there exist a subsequence $(y_n)$ of $(x_n)$
and a constant $C\geqslant 1$ such that inequality (\ref{e1}) is valid
provided that we restrict our attention to certain sequences of
coefficients (or if we project to certain finite subsets of $\nn$);
i.e. the sequence $(y_n)$ is \textit{partially} unconditional.

By now several partial unconditionality results have been obtained;
see, for instance, \cite{AGR,AT,DOSZ,LTo,Od} and the references therein.
The first (and undoubtedly the most influential) one was discovered by
J. Elton. To state it let us recall the following notion.
\begin{defn} \label{d3}
Let $0<\delta\leqslant 1$ and $K\geqslant 1$. A normalized basic sequence $(x_n)$ is said to be \emph{$\delta$-near unconditional
with constant $K$} if the basis constant of $(x_n)$ is at most $K$ and is such that for every $k\in\nn$, every $a_0,...,a_k\in\rr$
and every $G\subseteq\big\{ i\in\{0,...,k\}: \delta\leqslant |a_i|\leqslant 1\big\}$ we have
\begin{equation} \label{e2}
\big\| \sum_{n\in G} a_n x_n \big\| \leqslant K \big\| \sum_{n=0}^k a_n x_n \big\|.
\end{equation}
\end{defn}
We can now state Elton's Theorem.
\begin{thm} \label{t4}
\emph{(J. Elton \cite{Elt1})}
Let $0<\delta\leqslant 1$. Then there exists a constant $K\geqslant 1$, possibly depending on $\delta$, such that
for every normalized weakly null sequence $(e_n)$ in a Banach space $E$ there exists a subsequence $(x_n)$ of $(e_n)$
such that $(x_n)$ is $\delta$-near unconditional with constant $K$.
\end{thm}
For each $0<\delta\leqslant 1$ let $K(\delta)$ be the infimum of the set
of real numbers $K$ such that every normalized weakly null sequence has
a $\delta$-near unconditional subsequence with constant $K$.
\begin{pro} \label{p2}
\emph{(Elton unconditionality constant -- S. J. Dilworth, E. Odell, Th. Schlumprecht and A. Zs\'{a}k
\cite{DOSZ})} Does there exist $M\geqslant 1$ such that $K(\delta)\leqslant M$ for every $0<\delta\leqslant
1$?
\end{pro}
It is known (see \cite{DKK,Od}) that there exists an absolute constant
$C$ such that
\[ K(\delta)\leqslant C\log(1/\delta)\]
for every $0<\delta\leqslant 1$. It is also known (see \cite{DOSZ}) that
$5/4\leqslant \sup\{K(\delta):0<\delta\leqslant 1\}$.

\section{Rosenthal basis problem}

A normalized basic sequence $(x_n)$ in a Banach space $X$ is said to
be \textit{perfectly homogeneous} if every normalized block sequence
$(v_n)$ of $(x_n)$ is equivalent to $(x_n)$. It is easy to see that
the standard unit vector bases of the classical sequence spaces $c_0$
and $\ell_p$ $(1\leqslant p<+\infty)$ are perfectly homogeneous. Remarkably,
the converse is also true.
\begin{thm} \label{t5}
\emph{(M. Zippin \cite{Zi})}
Every perfectly homogeneous basic sequence $(x_n)$ is equivalent to the
standard unit vector basis of $c_0$ or $\ell_p$ for some $1\leqslant p<+\infty$.
\end{thm}
There is a slightly weaker notion: a normalized basic sequence $(x_n)$ in a Banach
space $X$ is said to be a \textit{Rosenthal basis} if every normalized block
sequence $(v_n)$ of $(x_n)$ has a subsequence which is equivalent to $(x_n)$.
\begin{pro} \label{p3}
\emph{(Rosenthal basis problem -- H. P. Rosenthal \cite{FPR})}
Is it true that every Rosenthal basis $(x_n)$ is equivalent to the standard
unit vector basis of $c_0$ or $\ell_p$ for some $1\leqslant p<+\infty$?
\end{pro}
It is known (see \cite{FPR}) that the problem has an affirmative answer provided
that there exists a constant $C\geqslant 1$ such that every normalized block
sequence $(v_n)$ of $(x_n)$ has a subsequence which is $C$-equivalent to $(x_n)$.

\section{Unique spreading model problem}

Let $(x_n)$ be a normalized basic sequence in a Banach space $X$ and $(y_n)$ be a sequence
in a Banach space $Y$. The sequence $(x_n)$ is said to \textit{generate the sequence $(y_n)$
as spreading model} if there exists a sequence $(\ee_n)$ of positive reals with $\ee_n\downarrow 0$
such that for every $n\in\nn$, every $a_0,...,a_n$ in $[-1,1]$ and every
$n\leqslant k_0< ... < k_n$ in $\nn$ we have
\[ (1+\ee_n)^{-1} \big\| \sum_{i=0}^n a_i y_i\big\| \leqslant
\big\| \sum_{i=0}^n a_i x_{k_i} \big\| \leqslant
(1+\ee_n) \big\| \sum_{i=0}^n a_i y_i\big\|.\]
It is well-known (see \cite{BrS1,BrS2}) that every normalized basic
sequence has a subsequence generating a spreading model. Actually, this
fact is one of the earliest applications of Ramsey Theory in the
Geometry of Banach spaces.

For every Banach space $X$ let $\mathrm{SP_w}(X)$ be the class of all
spreading models generated by a normalized weakly null sequence in $X$
and denote by $\sim$ the usual equivalence relation of equivalence of
basic sequences.
\begin{pro} \label{p4}
\emph{(Unique spreading model problem -- S. A. Argyros \cite{AOST})}
Let $X$ be a reflexive Banach space such that $|\mathrm{SP_w}(X)/\sim|=1$ (i.e. all
normalized weakly null sequences in $X$ generate a unique spreading model
up to equivalence). Is it true that the unique spreading model is equivalent
to the standard unit vector basis of $c_0$ or $\ell_p$ for some
$1\leqslant p<+\infty$?
\end{pro}
It is known (see \cite{DOS}) that the reflexivity assumption is essential.
It is also known (and easy to see) that an affirmative answer to
Problem \ref{p4} yields an affirmative answer to Problem \ref{p3}.

The following stronger version of Problem \ref{p4} has been also considered.
\begin{pro} \label{p5}
\emph{(Countable spreading models problem -- S. J. Dilworth, E. Odell and B. Sari \cite{DOS})}
Let $X$ be a reflexive Banach space such that $|\mathrm{SP_w}(X)/\sim|\leqslant \aleph_0$.
Does there exist a normalized weakly null sequence in $X$ generating a spreading model
equivalent to the standard unit vector basis of $c_0$ or  $\ell_p$ for some
$1\leqslant p<+\infty$?
\end{pro}
Some partial answers have been obtained. For instance, it is known (see \cite{DOS})
that if $X$ is reflexive, $|\mathrm{SP_w}(X)/\sim|\leqslant \aleph_0$ and
$|\mathrm{SP_w}(Y^*)/\sim|\leqslant \aleph_0$ for every infinite-dimensional subspace
$Y$ of $X$, then there exists a normalized weakly null sequence in $X$ generating a spreading model
equivalent to the standard unit vector basis of $c_0$ or $\ell_p$ for some
$1\leqslant p<+\infty$.

\section{Block homogeneous basis problem}

A normalized Schauder basis $(x_n)$ of a Banach space $X$ is said to be
\textit{block homogeneous} if every normalized block sequence $(v_n)$
of $(x_n)$ spans a subspace isomorphic to $X$.
\begin{pro} \label{p6}
\emph{(Block homogeneous basis problem -- see \cite{FR})}
Let $X$ be a Banach space with a normalized block homogeneous Schauder
basis. Is it true that $X$ is isomorphic to $c_0$ or $\ell_p$ for some
$1\leqslant p<+\infty$?
\end{pro}
Notice that an affirmative answer to the above problem would considerably
strengthen Theorem \ref{t5} and, combined with the deep results of
A. Szankowski \cite{Szankowski}, would yield an alternative proof
to the \textit{Homogeneous Banach Space Problem} that avoids the use of
W. T. Gowers' dichotomy \cite{G2}.

Although it is known that every separable non-Hilbertian Banach space contains
two non-isomorphic subspaces, we point out that there are several
related open problems. The following one is probably the most natural.
\begin{pro} \label{p7}
\emph{(G. Godefroy \cite{Go})}
Let $X$ be a separable non-Hilbertian Banach space. Is it true that there
exists a sequence $(X_n)$ of subspaces of $X$ which are pairwise non-isomorphic?
\end{pro}
It is likely that Problem \ref{p7} has an affirmative answer. Indeed, several
partial results have been obtained (see \cite{FR} and the references therein).

We would like to isolate an instance of Problem \ref{p7} exposing, in particular,
its connection with the \textit{Basis Problem}, a famous problem in the Geometry
of Banach spaces that has received considerable attention during the 1970s and 1980s.
Recall that, by the results in \cite{MP,Szankowski}, if a separable Banach space
$X$ fails to have type $2-\ee$ (or fails to have co-type $2+\ee$) for some
$\ee>0$, then $X$ contains a subspace without a Schauder basis (actually,
$X$ contains a subspace without the compact approximation property).
\begin{pro} \label{p8}
Let $X$ be a separable Banach space which fails to have type $2-\ee$ (or fails
to have co-type $2+\ee$) for some $\ee>0$. Is it true that $X$ contains
continuum many pairwise non-isomorphic subspaces?
\end{pro}
It is known (see \cite{Ani}) that the above problem has an affirmative answer
if $X$ is a separable weak Hilbert space not isomorphic to $\ell_2$.

\section{Fixing properties of operators on $C[0,1]$}

A central open problem in Banach Space Theory is the classification, up to isomorphism,
of all complemented subspaces of the classical function space $C[0,1]$.
\begin{pro} \label{csc}
\emph{(Complemented Subspace Problem for $C[0,1]$ -- see \cite{BP-new,LW-new,Pe1})}
Is it true that every complemented subspace of $C[0,1]$ is isomorphic to a $C(K)$ space
for some closed subset $K$ of $[0,1]$.
\end{pro}
The \textit{Complemented Subspace Problem} is discussed in detail in
\cite{Ro2}. The strongest result to date is due to H. P. Rosenthal
\cite{Ro1} and asserts that every complemented subspace of $C[0,1]$ with
non-separable dual is isomorphic to $C[0,1]$. Actually, this fact is a
consequence of the following surprising discovery concerning general
operators on $C[0,1]$.
\begin{thm} \label{t6}
\emph{(H. P. Rosenthal \cite{Ro1})}
Let $T:C[0,1]\to C[0,1]$ be a bounded linear operator such that its dual $T^*$
has non-separable range. Then $T$ fixes a copy of $C[0,1]$; that is, there
exists a subspace $Y$ of $C[0,1]$ which is isomorphic to $C[0,1]$ and is such
that $T|_Y$ is an isomorphic embedding.
\end{thm}
The following problem, asked in the 1970s, is motivated by the
\textit{Complemented Subspace Problem} and Theorem \ref{t6}.
\begin{pro} \label{p9}
\emph{(Fixing properties of operators on $C[0,1]$ -- H. P. Rosenthal \cite{Ro2})}
Let $T:C[0,1]\to C[0,1]$ be a bounded linear operator and $X$ be a separable Banach
space not containing a copy $c_0$. Assume that the operator $T$ fixes a copy of $X$.
Is it true that $T$ fixes a copy of $C[0,1]$? Equivalently, is it true that the dual
operator $T^*$ of $T$ has non-separable range?
\end{pro}
Notice that an affirmative answer to Problem \ref{p9} yields that every
complemented subspace $X$ of $C[0,1]$ with separable dual is hereditarily $c_0$
(that is, every infinite-dimensional subspace of $X$ contains a copy of $c_0$).
This property is predicted by the \textit{Complemented Subspace Problem}
(see \cite{PS}) though it has not been decided yet. It is known, however,
that every complemented subspace of $C[0,1]$ contains a copy $c_0$ (see \cite{Pe1}).

There are some partial positive answers to Problem \ref{p9}.
\begin{thm} \label{tB-new}
\emph{(J. Bourgain \cite{Bou})}
If an operator $T:C[0,1]\to C[0,1]$ fixes a copy of a co-type space, then $T$ fixes a copy of $C[0,1]$
\end{thm}
\begin{thm} \label{tG-new}
\emph{(I. Gasparis \cite{Ga})}
If an operator $T:C[0,1]\to C[0,1]$ fixes a copy of an asymptotic $\ell_1$ space, then $T$ fixes a copy of $C[0,1]$.
\end{thm}
Also, recently, a trichotomy was obtained characterizing, by means of fixing properties, the
class of operators on general separable Banach spaces whose dual has non-separable range.
To state it let us say that a sequence $(x_t)_{t\in\ct}$ in a Banach space $X$ is said
to be a \textit{tree basis} provided that the following are satisfied.
\begin{enumerate}
\item[(1)] If $(t_n)$ is the canonical enumeration\footnote{This means that for every $n,m\in\nn$
we have $n<m$ if and only if either $|t_n|<|t_m|$ or $|t_n|=|t_m|$ and $t_n<_{\mathrm{lex}} t_m$.}
of $\ct$, then $(x_{t_n})$ is a seminormalized basic sequence.
\item[(2)] For every infinite antichain $A$ of $\ct$ the sequence $(x_t)_{t\in A}$ is weakly null.
\item[(3)] For every $\sigma\in 2^\nn$ the sequence $(x_{\sigma|n})$ is weak* convergent to an element
$x^{**}_\sigma\in X^{**}\setminus X$. Moreover, if $\sigma, \tau\in 2^\nn$ with $\sigma\neq \tau$,
then $x^{**}_\sigma\neq x^{**}_\tau$.
\end{enumerate}
The archetypical example of such a sequence is the standard unit vector basis
of James tree space (see \cite{Ja-new}).
\begin{thm} \label{tD-new}
\emph{(P. Dodos \cite{D})}
Let $X$ and $Y$ be separable Banach spaces. Also let $T:X\to Y$ be an operator.
\begin{enumerate}
\item[(a)] If $X$ does not contain a copy of $\ell_1$, then $T^*$ has non-separable range
if and only if there exists a sequence $(x_t)_{t\in\ct}$ in $X$ such that both $(x_t)_{t\in\ct}$
and $(T(x_t))_{t\in\ct}$ are tree bases.
\item[(b)] If $X$ contains a copy of $\ell_1$, then $T^*$ has non-separable range if and only if
either the operator $T$ fixes a copy of $\ell_1$, or there exists a bounded sequence
$(x_t)_{t\in\ct}$ in $X$ such that its image $(T(x_t))_{t\in\ct}$ is a tree basis.
\end{enumerate}
\end{thm}
The proofs of the aforementioned results have a strong combinatorial flavor. In particular,
in \cite{Ga} heavy use is made of the infinite-dimensional Ramsey Theorem for
Borel partitions (see \cite{GP}) while the main result in \cite{D} is based on
the deep Halpern-L\"{a}uchli Theorem and its consequences (see \cite{HL,Ka,To2}).

\section{Bounded distortion}

Let $(X, \|\cdot\|)$ be an infinite-dimensional Banach space and $\lambda>1$.
The space $X$ is said to be \textit{$\lambda$-distortable} if there exists an
equivalent norm $|\cdot |$ on $X$ such that for every infinite-dimensional
subspace $Y$ of $X$ we have
\[ \sup\Big\{ \frac{|y|}{|z|}: y, z \in Y \text{ and } \|y\|=\|z\|=1\Big\} \geqslant \lambda.\]
The space $X$ is said to be \textit{distortable} if it is $\lambda$-distortable
for some $\lambda>1$. A distortable Banach space $X$ is said to be
\textit{arbitrarily distortable} if it is $\lambda$-distortable for
every $\lambda>1$; otherwise, $X$ is said to be \textit{boundedly distortable}.

The first result on distortion is due to R. C. James \cite{Ja}: \textit{$c_0$
and $\ell_1$ are not distortable}. V. D. Milman \cite{Mi} proved that
a non-distortable Banach space must contain almost isometric copies of
$c_0$ or $\ell_p$ for some $1\leqslant p <+\infty$. The first example of a
distortable Banach space was Tsirelson's space $T$ \cite{Ts}. The natural
problem whether the spaces $\ell_p$ $(1<p<+\infty)$ are distortable
became known as the \textit{Distortion Problem} and it was solved by
the following famous result.
\begin{thm} \label{t7}
\emph{(E. Odell and Th. Schlumprecht \cite{OS})}
For any $1<p<+\infty$ the space $\ell_p$ is arbitrarily distortable.
\end{thm}
The \textit{Distortion Problem} is discussed in detail in \cite{OS-new}.
A surprising phenomenon observed so far is that ``most" distortable
Banach spaces are arbitrarily distortable.
\begin{pro} \label{p10}
\emph{(Bounded distortion -- see \cite{OS-new})} Does there exist a boundedly distortable
Banach space?
\end{pro}
Tsirelson's space is a candidate for being boundedly distortable.
Indeed, we have already mentioned that $T$ is distortable;
actually, $T$ is $(2-\ee)$-distortable for every $\ee>0$ (see \cite{BL}).
But it not known whether $T$ is arbitrarily distortable.
\begin{pro} \label{p11}
\emph{(Distortion of Tsirelson's space -- H. P. Rosenthal)}
Is $T$ (or its dual) boundedly distortable?
\end{pro}
The \textit{Distortion Problem} has a non-linear analogue which
is also closely related to Ramsey Theory. Recall that a real-valued
Lipschitz map $f$ defined on the unit sphere $S_X$ of an infinite-dimensional
Banach space $X$ is said to be \textit{oscillation stable} if for every
$\ee>0$ there exists an infinite-dimensional subspace $Y$ of $X$ such that
the oscillation of $f$ on $S_Y$ is at most $\ee$, i.e.
\[ \sup\{ |f(y)-f(z)|: y, z\in Y \text{ and } \|y\|=\|z\|=1\}\leqslant \ee.\]
Notice that if a Banach space $X$ has the property that every real-valued
Lipschitz map $f$ on $S_X$ is oscillation stable, then $X$ is not distortable.
This is, however, a stronger property and is possessed only by hereditarily
$c_0$ spaces (see \cite{BL}). That the space $c_0$ has this property follows
by the following remarkable result.
\begin{thm} \label{t8}
\emph{(W. T. Gowers \cite{G1})} Every real-valued uniformly continuous
map on the unit sphere of $c_0$ is oscillation stable.
\end{thm}
All known proofs of Theorem \ref{t8} use methods of topological dynamics
on $\beta\nn$. It would be very interesting to have a purely combinatorial
proof avoiding the use of ultrafilters.

\section{Converse Aharoni problem}

The space $c_0$ is admittedly ``small" in the linear category: \textit{if
a Banach space $X$ is isomorphic to a subspace of $c_0$, then $X$ contains
a copy of $c_0$}. The situation, however, is dramatically different in
the non-linear category.
\begin{thm} \label{t9}
\emph{(I. Aharoni \cite{Ah})} Every separable metric space is Lipschitz
isomorphic to a subset of $c_0$.
\end{thm}
The best constant of the Lipschitz embedding has been computed (see \cite{KL}).
Theorem \ref{t9} can be rephrased to say that $c_0$ is universal in the
non-linear category. But it is not known whether $c_0$ is the
\textit{minimal} Banach space with this property.
\begin{pro} \label{p12}
\emph{(Converse Aharoni problem -- see \cite{BL,Kalton})}
Let $X$ be a separable Banach space with the property that every separable
Banach space is Lipschitz isomorphic to a subset of $X$. Is it
true that $X$ contains a copy of $c_0$?
\end{pro}
It is known (see \cite{GKL}) that the class of subspaces of $c_0$ is stable
under Lipschitz isomorphisms (moreover, if a Banach space $X$ is Lipschitz
isomorphic to $c_0$, then $X$ is linearly isomorphic to $c_0$). It is
possible that Theorem \ref{t8} could be used to attack Problem \ref{p12}.

\section{Separable quotient problem}

In the last four sections we will present some problems about general Banach spaces
where we do not assume separability. It turns out that this area has its own Ramsey
Theory to which it is related, the rich area of Ramsey Theory of uncountable structure
with a strong set-theoretic flavor (see, for example, \cite{To0,To5,To6}).

Recall the very old result due to S. Banach and S. Mazur saying that every
infinite-dimensional Banach space has an infinite basic sequence. It turns out
that the dual version of this \textit{Basic Sequence Problem} is still widely open.
\begin{pro} \label{p13}
\emph{(S. Banach \cite{B} -- A. Pe{\l}czy\'{n}ski \cite{Pe0})}
Does every infinite-dimensional Banach space $X$ has an infinite-dimensional
quotient with a Schauder basis?
\end{pro}
The following result represents the first major progress made on this problem.
\begin{thm} \label{t10}
\emph{(W. B. Johnson and H. P. Rosenthal \cite{JR})}
Every infinite-dimensional separable space $X$, and in fact every infinite-dimensional
space of density $<\mathfrak{b},$\footnote{Recall that $\mathfrak{b}$ is the minimal
cardinality of a subset of $\nn^\nn$ which is unbounded in the ordering of eventual dominance.}
has an infinite-dimensional quotient with a Schauder basis.
\end{thm}
So the problem of S. Banach and A. Pe{\l}czy\'{n}ski now becomes the following.
\begin{pro} \label{p14}
\emph{(Separable quotient problem)} Does every infinite-dimensional Banach
space have a non-trivial infinite-dimensional separable quotient?
\end{pro}
There are several partial results worth mentioning (see \cite{Mu} for a rather
complete historical review of the early work on this problem).
\begin{thm} \label{t11}
\emph{(W. B. Johnson and H. P. Rosenthal \cite{JR} -- J. N. Hagler and W. B. Johnson \cite{HJ})}
If $X^*$ contains an infinite-dimensional subspace with separable dual,
then $X$ has a non-trivial separable quotient. If $X^*$ has an unconditional
basic sequence, then $X$ has a quotient with an unconditional basis.
\end{thm}
\begin{thm} \label{t12}
\emph{(S. A. Argyros, P. Dodos and V. Kanellopoulos \cite{ADK1,ADK2})}
Every dual Banach space and every representable Banach space has a separable
non-trivial quotient.
\end{thm}
Concerning the two sufficient conditions given in Theorem \ref{t11} above,
we mention that W. T. Gowers \cite{G-new} has constructed a \emph{separable}
dual space which contains neither a copy of $\ell_1$ nor an infinite-dimensional
subspace with a separable dual.

\section{Quotients with long Schauder bases and biorthogonal systems}

Note that Problem \ref{p13} has also the following natural variant.
\begin{pro} \label{p15}
Given an infinite-dimensional Banach space $X$, what is the longest Schauder basis
a quotient of $X$ can have?
\end{pro}
Note that this in particular asks if every non-separable Banach space has an
uncountable biorthogonal system, a problem that has been first explicitly
addressed by W. J. Davis and W. B. Johnson \cite{DJ}. Today we know that the
problem is really about additional set-theoretic methods that can be relevant
in building uncountable biorthogonal systems and quotient maps. The first
and today well-known examples showing this were constructed by K. Kunen (see
\cite{Ku1,Ku2,N}) assuming CH and S. Shelah \cite{Sh2} assuming $\diamondsuit$.
Recently a systematic study of generic Banach spaces is given in \cite{LTo2}
and many of these examples are related to the uncountable biorthogonal system
problem. The following old example (see \cite[Chapter 1]{To0}) is not that
well-known and we mention it here because of its relevance to the discussion below.
\begin{thm} \label{t13}
\emph{(S. Todorcevic \cite{To0})}
If $\mathfrak{b}=\aleph_1$, then there is an Asplund $C(K)$ space without uncountable
biorthogonal systems.
\end{thm}
The second version of Problem \ref{p13} (Problem \ref{p15}) was analyzed in \cite{To1}
where the problem was connected with the following well-known set-theoretic and
Ramsey-theoretic principle.
\medskip

\noindent \textbf{P-ideal dichotomy (PID)}: For every P-ideal\footnote{Recall that
an ideal $\mathcal{I}$ is a P-ideal if for every sequence $(X_n)$ of elements of
$\mathcal{I}$ there is $Y$ such that $X_n\setminus Y$ is finite for all $n$.}
$\mathcal{I}$ consisting of countable subsets of some index-set $S$, either
\begin{enumerate}
\item[(1)] there is an uncountable set $T\subseteq S$ such that every countable
subset of $T$ belongs to $\mathcal{I}$, or
\item[(2)] the set $S$ can be decomposed into countably many subsets that have
no infinite subsets belonging to $\mathcal{I}$.
\end{enumerate}
\begin{thm} \label{t14}
\emph{(S. Todorcevic \cite{To1})} Assume {\rm PID}. Every nonseparable Banach space
$X$ of density $<\mathfrak{m}$ has a quotient with a Schauder basis of length $\omega_1$.
\end{thm}
Recall that $\mathfrak{m}$ is the minimal cardinality of a family $\mathcal{F}$ of
nowhere dense subsets of a compact ccc space $K$ such that $\bigcup\mathcal{F}=K$
(see, for example, \cite{Fr1}).
\begin{coro} \label{c15}
\emph{(S. Todorcevic \cite{To1})}
Assume {\rm PID}. If $\mathfrak{m}>\aleph_1$\footnote{Under {\rm PID}, the statement
$\mathfrak{m}>\aleph_1$ and the statement $\mathfrak{m}=\aleph_2$ are equivalent.
This holds also for other cardinals such as $\mathfrak{b}$ and $\mathfrak{p}$
(see \cite{To3,To4}).}, then every non-separable Banach space has an uncountable
biorthogonal system.
\end{coro}
The reader is referred to \cite{To3} and \cite{To4} which list known applications
of this principle many of which  have a strong Ramsey-theoretic flavor. It turns
out that {\rm PID} is an additional set-theoretic principle which, while independent
from CH, has a power of reducing problems about general sets to problems about reals.
Finding out what is the actual problem about reals that corresponds to the quotient
basis problem or the biorthogonal sequence problem is now in order. The following
is a particular instance of what one may expect from such analysis.
\begin{pro} \label{p16}
Are the following equivalent assuming the P-ideal dichotomy?
\begin{enumerate}
\item[(1)] Every non-separable Banach space has an uncountable biorthogonal system.
\item[(2)] $\mathfrak{b}=\aleph_2$.
\end{enumerate}
\end{pro}

\section{Rolewicz's problem on support sets}

Recall that a \emph{support set} in a Banach space $X$ is a nonempty convex set
$C$ with the property that every point $x$ of $C$ is its support point, i.e. there
is a functional $f$ of $X$ such that
\[ f(x)=\inf\{f(y): y\in C\}<\sup\{f(y): y\in C\}. \]
The existence of a support set in every non-separable Banach space is an old problem
of S. Rolewicz \cite{Rol} who showed that separable Banach spaces do not admit
support sets and who noticed that many of the non-separable Banach spaces
do admit support sets.
\begin{pro} \label{p17}
\emph{(S. Rolewicz \cite{Rol})}
Which Banach spaces admit support sets? Do all non-separable Banach spaces
have support sets?
\end{pro}
It is easily seen that if $X$ has an uncountable biorthogonal system, then $X$ has a support set, so the second part of this
question has a positive answer assuming $\mathrm{PID}$ and $\mathfrak{m}>\aleph_1$. It is also known that some assumptions
are necessary as it is consistence to have a $C(K)$ space of density $\aleph_1$ without support sets (see \cite{LTo2} and
\cite{Kos}). It turns out, however, that finding uncountable biorthogonal systems is a different problem from finding
support sets in Banach spaces.
\begin{thm} \label{t16}
\emph{(S. Todorcevic \cite{To1})} Every $C(K)$ space of density $>\aleph_1$ has a support set.
\end{thm}
Combining this with the following result we see the difference between
the Rolewicz problem and the uncountable biorthogonal sequence problem.
\begin{thm} \label{t17}
\emph{(C. Brech and P. Koszmider \cite{BK})}
It is consistent to have $C(K)$ spaces of density $>\aleph_1$ without
uncountable biorthogonal systems.
\end{thm}
This leaves however open the following interesting variant of Rolewicz's problem.
\begin{pro} \label{p18}
\emph{(S. Todorcevic \cite{To1})} Does every Banach space of density $>\aleph_1$
have a support set?
\end{pro}

\section{Unconditional basic sequences in non-separable spaces}

We now turn our attention to another well-known problem in this area.
\begin{pro} \label{p19}
\emph{(Unconditional basic sequences in non-separable spaces)}
Does every Banach space $X$ of density $>\mathfrak{c}$ have an infinite unconditional basic sequence? Does additional properties,
like reflexivity, make a difference?
\end{pro}
We mention some results related to this problem.
\begin{thm} \label{t18}
\emph{(S. A. Argyros, A. D. Arvanitakis and A. G. Tolias \cite{AAT})}
There is a Banach space $X$ of density $\mathfrak c$ with no infinite unconditional sequences (in fact, $X$ is hereditarily indecomposable)
nor an infinite-dimensional reflexive subspace.
\end{thm}
\begin{thm} \label{t19}
\emph{(S. A. Argyros, J. Lopez-Abad and S. Todorcevic \cite{ALT})}
There exists a reflexive Banach space of density $\aleph_1$ without infinite unconditional sequences.
\end{thm}
So, the unconditional basic sequence problem might have a different answers for different classes of Banach spaces. This has been hinted also
in a recent article by the authors where the following result is proven.
\begin{thm} \label{t20}
\emph{(P. Dodos, J. Lopez-Abad and S. Todorcevic \cite{DLT})}
It is consistent to assume that every Banach space of density $\geqslant \aleph_\om$ has an infinite unconditional sequence.
\end{thm}
\begin{coro} \label{c21}
\emph{(P. Dodos, J. Lopez-Abad and S. Todorcevic \cite{DLT})}
It is consistent to assume that every Banach space of density $\geqslant \aleph_\om$ has a quotient with an unconditional basis.
\end{coro}
There are related questions about the existence of infinite sub-symmetric sequences which seem to call for a different approach.
First of all let us recall the following well-known result of J. Ketonen in a slightly different form.
\begin{thm} \label{t22}
\emph{(J. Ketonen \cite{Ke})} Every Banach space of density at least the first Erd\H{o}s cardinal\footnote[5]{Recall that $\theta$
is said to be an \textit{Erd\H{o}s cardinal} if for every function $f:[\theta]^{<\omega}\to\omega$ there is an infinite set
$\Gamma\subseteq\theta$ such that $f$ is constant on $[\Gamma]^n$ for every $n<\omega$. This is a large cardinal which is larger
than the first inaccessible cardinal and, in particular, much larger than the continuum.} has an infinite sub-symmetric sequence.
\end{thm}
On the other hand no consistent bounds, like the one of \cite{DLT}
mentioned above, is known in this context. So the following problem is wide open.
\begin{pro} \label{p20}
Does every (respectively, every reflexive) Banach space of density $>\mathfrak{c}$
(respectively, of density $>\aleph_1$) has an infinite sub-symmetric basic
sequence?
\end{pro}
While analyzing this problem one is naturally led to the following variation.
\begin{pro} \label{p21}
Is there a non-separable Tsirelson-like space?
\end{pro}
This problem seems related to the existence of a nonseparable analogue of
the Schreier family on which the Tsirelson construction could be based.
This is of course related to the following well-known problem.
\begin{pro} \label{p23}
\emph{(D. H. Fremlin \cite{Fr2})}
Is there a compact hereditary family of finite subsets of $\omega_1$ containing
the singletons and having the property that every finite subset of $\omega_1$
has a subset half of its size belonging to the family $\mathcal{F}$?
\end{pro}

\noindent \textbf{Acknowledgment.} We would like to thank Gilles Godefroy
for his valuable comments and remarks.


text:
\chapter{Summary of results and future prospects}\label{conclusions}

In the present thesis, the behavior of the interacting Brownian motions and Bessel processes in the steady state and freezing regimes was investigated through the use of Dunkl operator theory. After the brief review of Dunkl theory given in Chapter~\ref{preliminaries}, the correspondence between the Calogero-Moser systems and Dunkl processes was proved in Chapter~\ref{CalogeroMoserCorrespondence} (Prop.~\ref{correspondencer}). This correspondence served as motivation for the fact that Dunkl processes, after given an appropriate scaling, converge to a steady state and have a well-defined freezing limit.

In Chapters~\ref{general_steady} and \ref{general_freezing}, the main results of this thesis were proved. The first result is that the scaled final distribution of a Dunkl process that starts from an initial distribution with finite second moments will converge to a specific steady-state distribution (Thm.~\ref{TheoremSteadyState}). The second result is that the scaled final distribution of a Dunkl process that starts from an arbitrary initial distribution freezes to a configuration that is given by the peak set of the root system $R$ instantaneously (Thm.~\ref{TheoremFreezingLimit}). The proof of these results depended on several calculations involving the intertwining operator, in particular the action of $V_\beta$ on linear polynomials (Lemma~\ref{LemmaFirstOrderV}), and on the exponential function in the freezing limit (Lemma~\ref{FreezingLimitDunklKernel}) as well as other approximations. While a finite lower bound was given for the time required for Dunkl processes to converge to the steady state, it seems that there must be a better estimation of the relaxation time in view of the fact that the freezing configuration is achieved instantaneously. This fact suggests that the relaxation time should be inversely proportional to the inverse temperature. This improvement on the estimation of the relaxation time is left as an open problem.

Because both the interacting Brownian motions and Bessel processes are particular cases of Dunkl processes, it follows from Thms.~\ref{TheoremSteadyState} and \ref{TheoremFreezingLimit} that these two systems of interacting particles have well-defined steady-state and freezing regimes. Chapter~\ref{ParticularCases} was devoted to these particular cases. The interacting Brownian motions converge to a steady state in which their scaled distribution coincides with the $\beta$-Hermite ensembles of random matrices, and freeze to a scaled distribution given by delta functions centered at the zeroes of the Hermite polynomials (Prop.~\ref{LimitingRegimesA}). Similarly, the interacting Bessel processes converge to a steady state in which their scaled distribution coincides with the $\beta$-Laguerre ensembles of random matrices, and freeze to a scaled distribution given by delta functions centered at the zeroes of the Laguerre polynomials; in addition, in the limit where the Bessel index tends to infinity (which, translated to the Dirac field of QCD corresponds to the case where the topological charge tends to infinity), all the particles converge to the same scaled position (Prop.~\ref{LimitingRegimesB}).

Prior to the derivation of Props.~\ref{LimitingRegimesA} and \ref{LimitingRegimesB}, the behavior of the interacting Brownian motions and Bessel processes in the steady-state and freezing regimes was studied using numerical simulations. As setup for the proof of these propositions, the action of $V_\beta$ on symmetric polynomials was derived in Prop.~\ref{PropositionVBetaOnSymmetricPolynomials}, and the freezing limit of the generalized Bessel functions of type $A$ and $B$ was obtained in Prop.~\ref{FreezingLimitOfGeneralizedBesselFunctions}. Because the expressions for the action of $V_\beta$ found in Chapter~\ref{ParticularCases} only apply to symmetric polynomials, it is of interest to examine its action on non-symmetric polynomials. This is a problem that should be tackled in the near future.

The general results in Thms.~\ref{TheoremSteadyState} and \ref{TheoremFreezingLimit} correspond to two regimes where the probability distribution of the process is balanced in such a way that the probability that the Dunkl process is in one particular Weyl chamber is evenly distributed among the chambers. This means that the process density is invariant under reflections along $\balpha\in R$, and the information about the jumps is lost. It is of great interest to study the physical nature of the jumps performed by Dunkl processes and their effect on the relaxation to the steady state, which is a problem that has not been solved yet. In particular, it is of interest to see if the behavior of the jumps in Dunkl processes has a relationship with a physical phenomenon.

In addition, the numerical results from Chapter~\ref{ParticularCases} (Figs.~\ref{FigureFreezingA} and \ref{FigureFreezingB} in particular) seem to suggest the existence of a transition from a disordered to an ordered phase as $\beta\to\infty$. However, in order to verify the existence of a phase transition the calculation of other physically relevant quantities (e.g., correlation functions) is required. Because the intertwining operator is responsible for the time evolution of Dunkl processes, the calculation of correlations in equilibrium should require different techniques from the ones used in this work. However, the results related to the intertwining operator obtained here open the possibility of studying dynamical and multi-time correlations for these processes. As a consequence, further study of the intertwining operator is essential to investigate whether these processes undergo phase transitions out of equilibrium. This is another topic that has not been addressed and that we would like to study as a continuation of this work.

Finally, it is worth noting that the interacting particle systems studied here seem to have little relationship with actual experiments. Because in the present most of the applications of random matrix theory correspond to ensembles where $\beta=2$, it is not unlikely that many other possible applications have been overlooked because the necessary tools for the study of cases where $\beta>0$ are incomplete. Hopefully, this work will be a stepping stone towards a better understanding of the models treated here for $\beta>0$, and towards finding out what makes the cases $\beta=2$ so special both in terms of their applications in physics and of their mathematical properties.


TITLE: Eigenvalues for an expression
QUESTION [1 upvotes]: While I was working I found a problem in which I encountered $$\det(A+xB)$$ where $A, B \in M_n(\mathbb{C}), x \in \mathbb{R}$. If we denote by $a_1, a_2, ..., a_n$ the eigenvalues of $A$ and by $b_1, b_2, ..., b_n$ the eigenvalues of $B$, it is known, for example, that $\det(A)=a_1a_2...a_n$. My question is:

Is it true that the eigenvalues of $A+xB$ are $(a_1+xb_1), (a_2+xb_2),...,(a_n+xb_n)$? Why? Does it hold for any kind of expression involving $2$ or more matrices?

For example, Lord Shark the Unknown used here that "only possible eigenvalues of $I+A-A^2$ are $1+0-0^2=1$ and $1+1-1^2=1$"

REPLY [3 votes]: Let $A = \begin{bmatrix} 2 & 1 \\ 1 & -2\end{bmatrix}$, then the trace is $0$ with determinant $-5$, hence the eigenvalues are $\sqrt5$ and $-\sqrt5$.
Let $B= \begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix}$, the trace is $0$ with determinant $1$, hence the eigenvalues are $i$ and $-i$.
If we add them together, we have $A+B=\begin{bmatrix} 2 & 2 \\ 0  &-2\end{bmatrix}$ with eigenvaluces $2$ and $-2$ which cannot be written as the sum of the eigenvalues of the eigenvalues of $A$ and $B$.

REPLY [1 votes]: This is not true in general, and it is very easy to write a counterexample. You should do this as an exercise.
However, if $A$ and $B$ have the same eigenspaces, associated with the corresponding indexes, then your formula holds. This is also easy to show. The converse case also holds because of  your variable $x$.


TITLE: How many natural solutions to $x_1+x_2+x_3+x_4=100$ with $x_1 \neq x_2 \neq x_3 \neq x_4$?
QUESTION [10 upvotes]: I have invented a little exercise of combinatorics that I can't solve (without brute forcing): in how many ways can I choose 4 non-negative integers, the sum of which is 100 and they are all different? So:
$x_1+x_2+x_3+x_4=100$
$x_1, x_2, x_3, x_4 \in \mathbb N$
$x_1 \neq x_2 \neq x_3 \neq x_4$
I have thought that the result have to be the number of total combination $\binom {100 + 4 - 1} {4 - 1}$ minus the ways I can solve $2x_1 + x_2 + x_3 = 100$. Anyway, I am not able to solve it either.
The solution should be:

 161664 or $\binom {100 + 4 - 1} {4 - 1} - 15187 = 176851 - 15187 = 161664$

Does anyone know how to calculate the combinations of this problem?

REPLY [2 votes]: Here is the solution from my side. One can translate this problem into a partition problem.
The much-studied partition function $p_k(n)$ is defined as the number of solutions of the equation $x_1+\cdots+x_k=n$ where $x_1\ge x_2\ge\cdots\ge x_k\ge1$; those solutions are called the partitions of $n$ into (exactly) $k$ parts. For fixed $k$, the generating function is
$$\sum_{k=0}^\infty p_k(n)x^n=\frac{x^k}{(1-x)(1-x^2)\cdots(1-x^k)}.$$
For $n\ge k\gt1$ we have the recurrence equation
$$p_k(n)=p_{k-1}(n-1)+p_k(n-k)$$
with initial values $p_0(0) = 1$ and $p_k(n) = 0$ if $n \le 0$ or $k \le 0$ , and $n$ and $k$ are not both zero
$x_1 + x_2 +x_3 +x_4 = 100$, where $x_i$ is non-negative integer.
Let, $x_1 = a_1$, $x_2 = x_1 + a_2$, $x_3 = x_2 + a_3$, $x_4 = x_3 + a_4$;
Where $a_1 \ge 0$ and $a_2, a_3, a_4 \ge 1$. This is to ensure that the numbers stay distinct
$4a_1 + 3a_2 + 2a_3 + a_4 = 100$
This is the exact same relation that appears when one resolves the equation of standard partition problem, except that $a_1$ should be $\ge 1$ and $a_2, a_3, a_4$ should be $\ge 0$.
We can do this by the following transformations
$a_1 = t_1 - 1$, $a_2 = t_2 + 1$, $a_3 = t_3 + 1$, $a_4 = t_4 + 1$
This satisfies our assumptions and now $t_1 \ge 1$ and $t_2, t_3, t_4 \ge 0$.
$4t_1 + 3t_2 + 2t_3 + t_4 = 98$
This is equivalent to dividing 98 into exactly 4 parts. Since all the 4 numbers can be aranged in any order to form other combinations of distinct non-negative $x_1, x_2, x_3, x_4$, therefore $4!$ should be multiplied.
The final answer would be $\Rightarrow 4! \cdot p_4(98)$
$p_4(98)$ can be calculated using the recurrence relation through hand or a computer program, but luckily we have closed form solutions available for certain values of $k$ in $p_k(n)$
$$p_4(n) = \left[ \frac{1}{144}t_4^3(n) - \frac{1}{48}t_4(n) \right]$$ for n is even
$$p_4(n) = \left[ \frac{1}{144}t_4^3(n) - \frac{1}{12}t_4(n) \right]$$ for n is odd
Here $\left[ \cdot \right]$ stands for nearest integer function and $t_k(n) = n + \frac{1}{4}.k(k-3)$
Subsitution values gives us $p_4(98) = \left[ \frac{99^3}{144} - \frac{99}{48} \right] = 6736$, $4! = 24$, $\Longrightarrow 24* 6736 = 161\:644$
A generalized approach for this problem has been disussed in this StackExchange question. This answer was also inspired by this answer


TITLE: Sanity check: Is every T-principal bundle over T trivial?
QUESTION [12 upvotes]: Is the following reasoning correct?

The classifying space of the 1-torus $\mathbb T$ is $\mathbb{CP^\infty}$.
Hence isomorphism classes of $\mathbb T$-principal bundles over $\mathbb T$ are in bijection with homotopy classes of continuous maps $\mathbb T\to\mathbb{CP^\infty}$, or equivalently, since $\mathbb{CP^\infty}$ is connected, with $\pi_1(\mathbb{CP^\infty})$.
But the $\mathbb T$-principal bundle $S^\infty\to\mathbb{CP^\infty}$ gives an exact (in the middle) sequence $\pi_1(S^\infty)\to\pi_1(\mathbb{CP^\infty})\to\pi_0(\mathbb T)$.
The groups $\pi_0(\mathbb T)$ and $\pi_1(S^\infty)$ vanish since $\mathbb T$ is connected and $S^\infty$ is contractible. Hence $\pi_1(\mathbb{CP^\infty})$ vanishes as well.
In conclusion, every $\mathbb T$-principal bundle over $\mathbb T$ is trivial.

This suprises me, because I feel like I could get a non-trivial bundle by "turning the fiber around once during one circulation."

REPLY [15 votes]: Your argument is fine - there is a unique principal $\mathbb{T}$ bundle over $\mathbb{T}$, namely, the trivial product.
Your intuition about turning the fiber around, I think, comes from confusing principal bundles with general bundles.
In fact, there are two fiber bundles over a circle whose fiber is a circle - the two dimensional torus and the Klein bottle (at least when the structure group is finite dimensional - though I believe the diffeomorphism group of $S^1$ retracts onto $O(2)$, so perhaps the finite dimensionality hypothesis is superfluous.)
In this case, one can prove that the structure group reduces down to $Gl(2,\mathbb{R})$.  Now, such a bundle over $S^1$ is trivial on the "northern" half and trivial on the "southern" half and must be glued together at the "east" and "west" poles (the points $(\pm 1,0)$ in $\mathbb{R}^2$).  The gluing is given by a (homotopy class of) map $\phi:S^0\rightarrow Gl(2,\mathbb{R})$ and we may assume wlog that $\phi$ of the basepoint of $S^0$ (whichever point we select to be the basepoint) is sent to the identity.  The map $\phi$ is known as the clutching function.
Then, because $GL(2,\mathbb{R})$ has 2 components (corresponding to those matrices of negative and positive determinant), there are exactly two homotopy class of such $\phi$ depending on which component $\phi$ of the non-basepoint is.
This proves there are two bundles over $S^1$ with fiber $S^1$.  It's not too hard to pictorially draw the nontrivial bundle.  Draw the Klein bottle as a square with sides identified.  Identify the upper side with lower orientation preservingly and identify the sides orientation reversingly.  Then the map $\pi$ defining the fiber bundle structure sends a point in the square to it's projection along the bottom line.  The fibers are the vertical lines (circles after identification).
So we have two bundles, the trivial bundle and the Klein bottle.  You have already proven that the trivial bundle is a principal bundle (and is the only one), but let me give some intuition for why that the Klein bottle is not.
Pick a fiber of $K$.  I'm thinking of the fiber going through the center of the square.  In order to be a principal bundle, $\mathbb{T}$ must act simply transitively on this fiber.  There are precisely two such actions, namely $z* w = zw$ or $z*w = \overline{z}w$.  Intuitively, these actions corresponding to the circle acting on itself by either spinning one direction or the opposite direction.  In our square model of $K$, this corresponds to moving points either up or down along the vertical line containing them.  I'll assume we're pick "moving up", the other case being basically identical.
By continuity of the action, nearby fibers must also move up.  In fact, one can extend this everywhere - by continuity, $\mathbb{T}$ must act on all the fibers by moving them up.  But if you now look at how the left edge and right edge are identified, you'll see you've now specified contradictory actions on the identified left/right edge.  This shows $K$ is not a principal bundle, even though it is a perfectly good fiber bundle.
The moral of the story is that "principal $G$-bundle" and "fiber bundle with fiber $G$" are different ideas.  The second needs a consistent choice of "multiplication by G" on all fibers in order to become the first and sometimes this consistent choice can't be made.


TITLE: Hales Jewett Theorem
QUESTION [4 upvotes]: In the book Ramsey Theory by Graham, Rothschild and Spencer the authors state:

The Hales Jewett Theorem strips van der Waerden's theorem of its unessential elements and reveals the heart of Ramsey theory. It provides a focal point from which many of the results can be derived and acts as a cornerstone of much of the more advanced work. Without this result Ramsey theory would more properly be called Ramseyian theorems.

I am interested in knowing as to what further advanced results flow from the Hales Jewett theorem and why is the theorem so crucial.
Thanks for your time.

REPLY [2 votes]: Here is an example of an easy corollary of the Hales-Jewett theorem:
Theorem: For every finite topological space $X$ and $r \in \mathbb{N}$, there is a finite topological space $Y$ such that if $Y$ is $r$-colored, then we can find a monochromatic copy of $X$.
Proof: Using the Hales-Jewett theorem, pick $n$ large enough so that any $r$-coloring of $X^n$ provides a monochromatic combinatorial line. Then set $Y = X^n$ (with the usual product topology), and observe that every combinatorial line in $X^n$ is homeomorphic to $X$.
This type of result fits into the area called induced Ramsey theory. The Hales-Jewett theorem gives us many such results: the same proof works for finite metric spaces (use the sup metric on the product), graphs (use the strong product), digraphs (use a suitable generalization of the strong product), and more. It works for finite groups too, provided you are willing to weaken the conclusion a bit:
Theorem: For every finite group $G$ and $r \in \mathbb{N}$, there is a finite group $H$ such that if $H$ is $r$-colored, then we can find a subgroup $G'$ of $H$ isomorphic to $G$ and an element $h \in H$ such that $hG$ is monochromatic.
Proof: Essentially the same: just observe that in $G^n$, combinatorial lines are shifted copies of $G$.
Of course, many results from induced Ramsey theory involve coloring not points, but pairs of points, triples of points, etc. These kinds of results do not seem to follow easily from the Hales-Jewett theorem.
In addition to these kinds of results, the Hales-Jewett theorem is used in the proof of the Graham-Leeb-Rothschild theorem, the edge-induced graph theorem, and its bipartite version (these are in sections 2.4, 5.2, and 5.3 of the Graham-Rothschild-Spencer book). It also gives quick proofs for van der Waerden's theorem and its higher-dimensional analogue, Gallai's theorem. See also Exercise 15 of this blog post of Terry Tao for further examples of what you can do using the Hales Jewett theorem.


TITLE: Triangle Inequality about complex numbers, special case
QUESTION [1 upvotes]: At Spivak's calculus he treats the triangle inequality about complex numbers dividing into cases. Let $z$ and $w$ complex numbers, the inequality is clear if $z$ or $w$ are $0$.
$\lvert z+w \rvert$$\leq \lvert z \rvert + \lvert w \rvert$
Suppose that $z=\lambda w$, then it is straightforward to prove the inequality but divide in cases :
$\lambda<0$  and $\lambda>0$.
I can't see how is this straightforward and why do we need to divide into cases for $\lambda$.
After I do the algebraic manipulations it seems that it doesn't matter how $\lambda$ is.
My guess is that my approach is wrong, so if someone can give me a hint on how to initiate the proof would be a great help. Thank you

REPLY [0 votes]: Let $z=(a_1,b_1), w=(a_2,b_2)$, suppose $z=lw$, then $z=(la_1,la_2)$
I want to prove $\lvert z+w \rvert \leq\lvert z\rvert+\lvert w\rvert$
calculate $\lvert z+w \rvert$ = $\bullet$:
$\bullet$=$\lvert (la_2,lb_2)+(a_2,b_2) \rvert$=$\lvert (la_2+a_2,lb_2+b_2)\rvert$= $\sqrt{(1+l)^2a^2+(1+l)^2b^2)}$=$\lvert 1+l \rvert$ $\sqrt{a^2+b^2}$
calculate $\lvert z \rvert+\lvert w \rvert$= $\star$ :
$\star$ =$\lvert (la_2,lb_2)\rvert$ + $\lvert (a_2,b_2) \rvert$= $\sqrt{l^2a^2+l^2b^2}$+$\sqrt{a^2+b^2}$=$\rvert l \lvert$ + 1 $\sqrt{a^2+b^2}$
if l>$0$ then $\lvert 1+l \rvert$=$\rvert l \lvert$ + 1 then $\lvert z+w \rvert =\lvert z\rvert+\lvert w\rvert$
if l


TITLE: Does dark matter exist or not?
QUESTION [0 upvotes]: I've seen articles in recent days (e.g. https://physics.aps.org/articles/v14/s71) about an experiment in Spain possibly disproving the precedent finding by DAMA/LIBRA.
At the same time there are new articles about discovering a "web of dark matter" connecting the galaxies.
Have we detected dark matter yes or no?

REPLY [4 votes]: The evidence for dark matter is now sufficiently varied and robust that there is not really any room for doubt that it does indeed exist in roughly spherical regions centred on galaxies and in filaments between galaxies. For a summary of the evidence see other questions on this site, and wikipedia, among other places.
The evidence is from the gravitational effects, and the way these influence other things such as the way galaxies formed in the first place. But no one has yet found a particle of dark matter in their laboratory experiment on Earth, detected through some other signature such as an interaction with other particles, or a decay or something like that.

REPLY [3 votes]: The article you linked merely provided counterevidence against one single direct dark matter detection done by the DAMA/LIBRA collaboration in Italy nearly 20 years ago. Nobody has ever found a reproducible result of direct dark matter detection. That Italian collaboration claimed to have found the expected signature of dark matter (DAMA/LIBRA wiki):

This experiment has further confirmed the presence of a
model-independent annual modulation effect in the data in 2-6 keV
range that satisfy all the features expected for a dark matter signal
with high statistical significance.

A large number of people working in the field direct-detection of dark matter were doubting the DAMA/LIBRA result, and now a separate experiment has finally refuted their claim (though not with total certainty just yet).
As for indirect detection of dark matter, there are gazillions of reproducible detections and measurements. Dark matter has irrefutably been indirectly detected, and a great deal of its properties have been measured. All of this information is detailed in the wikipedia article.
The situation is, through various indirect measurements we know that dark matter exists and it has certain properties, but nobody knows what it really is, i.e. what it is fundamentally comprised of, so to speak.


TITLE: $x^2 \equiv 2x \pmod m$
QUESTION [4 upvotes]: Toward counting the solutions for the congruence $x^2 \equiv 2x \pmod n$, if we write $m$
 as $m = p_1^{a_1}p_2^{a_2}...p_r^{a_r}$
we have the following equivalent system of congruence equations:
$p_1^{a_1}\mid x(x-2)$
$p_2^{a_2}\mid x(x-2)$
.
.
.
$p_r^{a_r}\mid x(x-2)$
I am thinking of the two cases of the relationships of $x$ and $x-2$ where

$gcd(x,x-2) = 1$
$gcd(x,x-2) = 2$

and consider the number of solutions for each case. Am I on the right track?

REPLY [4 votes]: Using your approach, let for prime $p,$ $p\mid x$ and $p\mid (x-2)$
$\implies p$ divides $x-(x-2)=2\implies p=2$ for $p^n\mid x(x-2)$
So, if prime $p>3,$ either $p^n\mid x$ or $p^n\mid (x-2)$
So, there are $2$ in-congruent solutions, namely, $0,2\pmod {p^n}$ of  $x^2\equiv2x \pmod {p^n}$.
If case $p=2,x$ must be even $\implies (x,x-2)=2\implies(\frac x2,\frac{x-2}2)=1$
So, $2^n\mid x(x-2)\implies 2^{n-2}\mid \frac x2\cdot\frac{(x-2)}2$ if $n\ge 3$
So,  either $2^{n-2}\mid \frac x2$ or $2^{n-2}\mid \frac{(x-2)}2$
So, $x\equiv0,2 \pmod{2^{n-1}}\implies x\equiv0,0+2^{n-1},2,2+2^{n-1}\pmod{2^n}$ i.e., there are $4$ in-congruent solutions.
If $n=2, x(x-2)\equiv 0\pmod4\implies x\equiv0,2\pmod4$ i.e., $2$ in-congruent solutions.
If $n=1, x(x-2)\equiv 1\pmod2\implies x\equiv 1\pmod2$  i.e., exactly $1$ in-congruent solution.
Now,  if $ax^2+bx+c\equiv 0\pmod {m_1}$ and $ax^2+bx+c\equiv 0\pmod {m_2}$ have $t_1,t_2$ solutions respectively(where $(m_1,m_2)=1$),
$ax^2+bx+c\equiv 0\pmod {m_1\cdot m_2}$ will have $t_1\cdot t_2$ solutions.

Alternatively,
$x^2\equiv2x\pmod n\implies (x-1)^2\equiv 1\pmod n\implies y^2\equiv1$ if $y=x-1$
If $y^2\equiv1 \pmod {q^n}$ where prime $q>2$
So, $q^n \mid (y-1)(y+1)$ but $(y-1,y+1)\mid 2$
either $q^n\mid (y-1)$ or $q^n\mid (y+1)$
So, there are $2$ in-congruent solutions, namely, $y\equiv \pm 1\pmod {q^n}$ of  $y^2\equiv1 \pmod {q^n}$.
If $y^2\equiv1 \pmod {2^n}$,
so, $2^n\mid (y-1)(y+1)$ ,here $(y-1,y+1)=2$ as $y$ is odd.
so, $2^{n-2} \mid\frac{(y-1)}2\cdot\frac{(y+1)}2$ where
$(\frac{y-1}2,\frac{y+1}2)=1$ if $n\ge 3$
so, either $2^{n-1}\mid (y-1)$ or $2^{n-1}\mid (y+1)$
So, there are $2$ in-congruent solutions, namely, $y\equiv \pm 1\pmod {2^{n-1}}$
So, there will be $4$ in-congruent solutions, namely, $y\equiv \pm 1,2^{n-1}\pm1\pmod {2^n}$
If $n=2, y^2\equiv 1\pmod4\implies y\equiv\pm 1\pmod4$ i.e., $2$ in-congruent solutions.
If $n=1, y^2\equiv 1\pmod2\implies y\equiv 1\pmod2$  i.e., exactly $1$ in-congruent solution.


TITLE: The Petersburg game and fair game (Feller Vol.1)
QUESTION [4 upvotes]: (Feller Vol.1, P.252, Petersburg's paradox) Let $(X_k)$ be mutually independent random variables, each of which assume $2, 2^2, 2^3, ...$ with probability $2^{-1}, 2^{-2}, 2^{-3}, ...$. We define $S_n = X_1 + \cdots + X_n$. Let $e_n$ be the accumulated entrance fees, and $S_n$ be the accumulated gains. The author says that the game is fair if for every $\epsilon >0$,
$$P(|\frac{S_n}{e_n} -1|> \epsilon) \to 0.$$
Then, he claims that in the Petersburg game, $e_n = n \log_2 (n)$ makes it fair. The proof is given as follows:
First, he introduces truncated variables $U_k$ and $V_k$. If $n \log_2(n) \ge X_k$, $U_k=X_k$, $V_k =0$. On the other hand, if $n \log_2(n) < X_k$, $U_k = 0$, $V_k = X_k$. Then, it follows  that
$$P(|\frac{S_n}{e_n} -1|> \epsilon) \le P(|U_1+ \cdots + U_n - e_n| > e_n \epsilon) + P(V_1 + \cdots + V_n \not=0).$$
The inequality holds since if neither of two events on the right does occur, the event on the left cannot happen. Thus, it is sufficient to prove that two probability on the right converges to $0$. The author uses the following inequality
$$P(V_1 + \cdots + V_n \not= 0) \le n \sum_{i=1}^n P(V_1 \not= 0) \le P(X_1 > n \log_2(n) ) \le \frac{2}{\log_2(n)} \to 0.$$
However, I don't understand how the last inequality holds. I was thinking about using Chebyshev's inequality, or the fact that $P(X_1 > n \log_2(n)) = P(X_1 \ge 2^{r+1})$ assuming that $r$ is the largest integer satisfying $2^{r} \le n \log_2(n)$. But, I failed both attempts.
I would appreciate if you give some help, and please let me know if you need more context.

REPLY [1 votes]: Your second suggestion works. Let $r$ be the greatest natural number satisfying $2^r \leq n \log n$. Then $X_1 > n \log n$ implies $X_1 \geq 2^{r+1}$. So,
$$P(X_1 > n \log n) \leq P(X_1 \geq 2^{r+1}) = \sum_{k=r+1}^\infty \frac{1}{2^k}= \frac{1}{2^r} <\frac{2}{n\log n},$$
where the last inequality is just rearranging $2^{r+1} > n \log n$.


TITLE: Equation with infinite set difference and intersection
QUESTION [0 upvotes]: Given  $$ \dots A_2\subseteq A_1\subseteq A_0 = U$$
and
$$\bigcup_{n=0}^\infty (A_n \backslash A_{n+1})\cup\left(\bigcap_{n=0}^\infty  A_n\right) = U$$
when $U$ is the universal set, prove that
$$(A_1 \backslash A_2)\cup(A_3 \backslash A_4)\cup \dots \cup\left(\bigcap_{n=0}^\infty A_n \right) = U\backslash \left( (A_0 \backslash A_1)\cup(A_2 \backslash A_3)\cup \dots \right)$$
My try:
I am almost sure that this problem can be solved with simple induction. The problem is i can't write the induction assumption like so:
$$(A_1 \backslash A_2)\cup(A_3 \backslash A_4)\cup \dots \cup (A_{k-2} \backslash A_{k-1})\cup \left(\bigcap_{n=0}^k A_n \right) = U\backslash \left( (A_0 \backslash A_1)\cup(A_2 \backslash A_3)\cup \dots \cup (A_{k-1} \backslash A_k)\right)$$
because:

I don't know if k is even or odd.
There are two cases. for example:
$$k=2 \to (A_1 \backslash A_2)\cup (A_0 \cap A_1 \cap A_2) = U \backslash (A_0 \backslash A_1)$$

$$k=3 \to (A_1 \backslash A_2)\cup (A_0 \cap A_1 \cap A_2 \cap A_3) = U \backslash ((A_0 \backslash A_1) \cup (A_2 \backslash A_3))$$
hence
$$k=2 \to (A_1 \backslash A_2)\cup A_2 = U \backslash (A_0 \backslash A_1)$$
$$k=3 \to (A_1 \backslash A_2)\cup A_3 = U \backslash ((A_0 \backslash A_1) \cup (A_2 \backslash A_3))$$
so by looking at the left term, the index of the intersection has changed but the set difference is still the same.
Can you please help me find the right way to induce this proof, or is there a better and simpler way i just can't see?

REPLY [1 votes]: HINT: For convenience let $D_n=A_n\setminus A_{n+1}$ for $n\in\Bbb N$, and let $C=\bigcap_{n\in\Bbb N}A_n$. Use the hypotheses to show that $\mathscr{P}=\{C\}\cup\{D_n:n\in\Bbb N\}$ is a family of pairwise disjoint sets whose union is $U$. (In other words, they form a partition of $U$ except that some may be empty.) A generalization of the desired result can then be proved by element-chasing: if $\mathscr{A}\subseteq\mathscr{P}$, then for each $x\in U$ we have $x\in\bigcup\mathscr{A}$ if and only if $x\notin\bigcup(\mathscr{P}\setminus\mathscr{A})$.
Induction isn’t appropriate here: you’re not trying to prove a family of statements, one for each $k\in\Bbb N$. Rather, you’re trying to prove a single statement that just happens to mention all $k\in\Bbb N$.