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Hindman (Furstenberg)




[latex]


\begin{theorem}[Hindman]


\label{thm:hindman}


\label{thm:hindmanfurstenberg}


If $\N$ is partitioned into finitely many


sets,


then there is is an infinite subset $H \subseteq \N$


such that all finite sums of distinct


elements of $H$


belong to the same set of the partition.


\end{theorem}


[/latex]


Use:


[latex]


\begin{theorem}


\label{thm:unifrprox}


Let $X$ be a compact Hausdorff space and $T\colon X \to X$


continuous.


Consider $(X,T)$.%TODO different notations


Then for every $x \in X$


there is a uniformly recurrent $y \in X$


such that $y $ is proximal to $x$.


\end{theorem}


[/latex]


[latex]


\begin{refproof}{thm:hindmanfurstenberg}[Furstenberg]


\begin{itemize}


\item View partition as $f\colon \N \to k$. Consider $X \coloneqq k^{\N}$ (product topology, compact and Hausdorff).


Let $x \in X$ be the given partition.


\item $T\colon X \to X$ shift: $T(y)(n) \coloneqq y(n+1)$.


\item Let $y$ proximal to $x$, uniformly recurrent.


\begin{itemize}


\item proximal $\leadsto$ $\forall N$.~$T^n(x)\defon_N = T^n(y)\defon_N$


for infinitely many $n$.


\item uniform recurrence $\leadsto$


\[


\forall n .~\exists N.~\forall r.y\defon{\{r,\ldots,r+N1\}}


\text{ contains } $y\defon{\{0,\ldots,n\}}$ \text{ as a subsequence.}


\]


(consider neighbourhood $G_n = \{z \in X : z\defon{n} = y\defon{n} \}$).


\end{itemize}


\item Consider $c \coloneqq y(0)$. This color works:


\begin{itemize}


\item $G_0 \coloneqq y\defon{\{0\}}$,


take $N_0$ such that $y\defon{\{r, \ldots, r + N_0  1\}} $


contains $y(0)$ for all $r$ (unif.~recurrence).


$y\defon{\{r,\ldots,r+N_0  1\} } = x\defon{\{r,\ldots,r+N_0 1\} }$


for infinitely many $r$ (proximality).


Fix $h_0 \in \N$ such that $x(h_0) = y(0)$.


\item $G_1 \coloneqq y\defon{\{0,\ldots,h_0\} }$,


take $N_1$ such that $y\defon{\{r,\ldots,r +N_11\}}$


contains $y\defon{\{0,\ldots,h_0\} }$


for all $r$ (unif.~recurrence).


So among ever $N_1$ terms, there are two of distance $h_0$


where $y$ has value $c$.


So $\exists h_1 > h_0$ such that $x(h_1) = x(h_1 + h_0) = c$


(proximality).




\item Repeat:


Choose $h_i$ such that


for all sums $s$ of subsets of $\{h_0,\ldots, h_{i1}\}$,


$x(s+h_i) = y(s+h_i) = c$:


Find $N_i$ such that every $N_i$ consecutive


terms of $y$ contain a segment that coincides


with the initial segment of $y$


up to the largest $s$,


then find a segment of length $N_i$ beyond $h_{i1}$


where $x$ and $y$ coincide.


\end{itemize}


\end{itemize}


\end{refproof}


[/latex]

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