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

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 r expansions, and their various generalisations: generalised Lüroth series expansions and β \beta -expansions. In fact, we consider subshifts over a countable alphabet generated by all possible expansions of numbers in [ 0 , 1 ) [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 Π 3 0 \boldsymbol {\Pi }^0_3 -complete set, meaning that it is a countable intersection of F σ F_\sigma -sets, but it is not possible to write it as a countable union of G δ 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 Π 3 0 \boldsymbol {\Pi }^0_3 -complete.