Borel complexity of sets of normal numbers via generic points in subshifts with specification

Airey, Dylan; Jackson, Steve; Kwietniak, Dominik; Mance, Bill

doi:10.1090/tran/8001

Borel complexity of sets of normal numbers via generic points in subshifts with specification
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by Dylan Airey, Steve Jackson, Dominik Kwietniak and Bill Mance PDF

Trans. Amer. Math. Soc. 373 (2020), 4561-4584 Request permission

Abstract:

We study the Borel complexity of sets of normal numbers in several numeration systems. Taking a dynamical point of view, we offer a unified treatment for continued fraction expansions and base $r$ expansions, and their various generalisations: generalised Lüroth series expansions and $\beta$-expansions. In fact, we consider subshifts over a countable alphabet generated by all possible expansions of numbers in $[0,1)$. Then normal numbers correspond to generic points of shift-invariant measures. It turns out that for these subshifts the set of generic points for a shift-invariant probability measure is precisely at the third level of the Borel hierarchy (it is a $\boldsymbol {\Pi }^0_3$-complete set, meaning that it is a countable intersection of $F_\sigma$-sets, but it is not possible to write it as a countable union of $G_\delta$-sets). We also solve a problem of Sharkovsky–Sivak on the Borel complexity of the basin of statistical attraction. The crucial dynamical feature we need is a feeble form of specification. All expansions named above generate subshifts with this property. Hence the sets of normal numbers under consideration are $\boldsymbol {\Pi }^0_3$-complete.

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