Let $\mathcal{A}$ be a finite set. Is there a nice characterization of the subset of $S\subset \mathcal{A}^\omega$ such that every $w\in S$ has finite critical exponent? Of course $S$ has measure zero (with respect to the uniform product measure), but what can be said about its topological properties?
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asked Feb 4, 2021 at 13:43
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1$\begingroup$ The set is also $F_\sigma$ and meager - for every $n$, the set of words which contain an $n$-th power is open and dense. $\endgroup$– WojowuCommented Feb 4, 2021 at 17:44
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$\begingroup$ That is correct. Indeed, the map $\kappa_n$ defined here: mathoverflow.net/q/376516/167834 is continuous on its zero set, which is thus $G_\delta$. Since it is also dense ($\kappa_n$ for instance vanishes on $\mathbb{Q}$), by Baire category theorem it is residual, so that its complement is $F_\sigma$ and meagre. $\endgroup$– Alessandro Della CorteCommented Feb 4, 2021 at 17:57
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$\begingroup$ What else can be said from a topological point of view? $\endgroup$– Alessandro Della CorteCommented Feb 4, 2021 at 18:00
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