75 lines
3 KiB
Text
75 lines
3 KiB
Text
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Hindman (Furstenberg)
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[latex]
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\begin{theorem}[Hindman]
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\label{thm:hindman}
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\label{thm:hindmanfurstenberg}
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If $\N$ is partitioned into finitely many
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sets,
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then there is is an infinite subset $H \subseteq \N$
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such that all finite sums of distinct
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elements of $H$
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belong to the same set of the partition.
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\end{theorem}
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[/latex]
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Use:
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[latex]
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\begin{theorem}
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\label{thm:unifrprox}
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Let $X$ be a compact Hausdorff space and $T\colon X \to X$
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continuous.
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Consider $(X,T)$.%TODO different notations
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Then for every $x \in X$
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there is a uniformly recurrent $y \in X$
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such that $y $ is proximal to $x$.
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\end{theorem}
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[/latex]
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[latex]
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\begin{refproof}{thm:hindmanfurstenberg}[Furstenberg]
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\begin{itemize}
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\item View partition as $f\colon \N \to k$. Consider $X \coloneqq k^{\N}$ (product topology, compact and Hausdorff).
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Let $x \in X$ be the given partition.
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\item $T\colon X \to X$ shift: $T(y)(n) \coloneqq y(n+1)$.
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\item Let $y$ proximal to $x$, uniformly recurrent.
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\begin{itemize}
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\item proximal $\leadsto$ $\forall N$.~$T^n(x)\defon_N = T^n(y)\defon_N$
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for infinitely many $n$.
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\item uniform recurrence $\leadsto$
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\[
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\forall n .~\exists N.~\forall r.y\defon{\{r,\ldots,r+N-1\}}
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\text{ contains } $y\defon{\{0,\ldots,n\}}$ \text{ as a subsequence.}
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\]
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(consider neighbourhood $G_n = \{z \in X : z\defon{n} = y\defon{n} \}$).
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\end{itemize}
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\item Consider $c \coloneqq y(0)$. This color works:
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\begin{itemize}
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\item $G_0 \coloneqq y\defon{\{0\}}$,
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take $N_0$ such that $y\defon{\{r, \ldots, r + N_0 - 1\}} $
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contains $y(0)$ for all $r$ (unif.~recurrence).
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$y\defon{\{r,\ldots,r+N_0 - 1\} } = x\defon{\{r,\ldots,r+N_0 -1\} }$
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for infinitely many $r$ (proximality).
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Fix $h_0 \in \N$ such that $x(h_0) = y(0)$.
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\item $G_1 \coloneqq y\defon{\{0,\ldots,h_0\} }$,
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take $N_1$ such that $y\defon{\{r,\ldots,r +N_1-1\}}$
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contains $y\defon{\{0,\ldots,h_0\} }$
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for all $r$ (unif.~recurrence).
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So among ever $N_1$ terms, there are two of distance $h_0$
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where $y$ has value $c$.
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So $\exists h_1 > h_0$ such that $x(h_1) = x(h_1 + h_0) = c$
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(proximality).
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\item Repeat:
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Choose $h_i$ such that
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for all sums $s$ of subsets of $\{h_0,\ldots, h_{i-1}\}$,
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$x(s+h_i) = y(s+h_i) = c$:
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Find $N_i$ such that every $N_i$ consecutive
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terms of $y$ contain a segment that coincides
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with the initial segment of $y$
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up to the largest $s$,
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then find a segment of length $N_i$ beyond $h_{i-1}$
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where $x$ and $y$ coincide.
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\end{itemize}
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\end{itemize}
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\end{refproof}
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[/latex]
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