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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]