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 expansions, and their various generalisations: generalised Lüroth series expansions and β-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 Π 0 3 -complete set, meaning that it is a countable intersection of Fσ-sets, but it is not possible to write it as a countable union of G δ -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 0 3 -complete.where [w] = {z ∈ A ω : z 0 = w 1 , . . . , z n−1 = w n }. By the shift-invariance of µ the measure of [w] is equal to the µ-probability of the occurrence of w at any fixed position ℓ ∈ ω, that is, µ([w]) = µ({z ∈ A ω : z ℓ = w 1 , . . . , z ℓ+n−1 = w n }).

Generic point equivalence and Pisot numbers

“…This simple case already seems new. Indeed, this serves an alternative proof of Corollary 19 in [1] which solves several conjectures posed in [23], as the set of 2-normal numbers lies exactly in the 3rd Borel-hierarchy by [9].…”

Some complexity results in the theory of normal numbers

“…We now return to the proof of Lemma 4.5. Given x ∈ 2 ω 3 , we define a function h(x)∶ ω 3 → ω 3 as follows. We let h(x)(i, j, 0) be (1, j + 1, 0) if x(i ′ , j, 0) = 0 for all i ′ ≤ i.…”

“…(3) b j > (j + 1) 2 2 j+1 B(δ j+1 , 1 ( j+1) 2 , f ( j + 1)), where we recall f ( j) = 2 j 2 , and B is as in Lemma 3.2. As in Lemma 4.2, we then let n i , j = a i b j , B i , j = [n i , j−1 , n i , j ).…”

On the Borel complexity of continued fraction normal, absolutely abnormal numbers

“…and define D 2 (Σ 0 α ) similarly. 2 The sets D 2 (Π 0 α ) and D 2 (Σ 0 α ) contain the sets Π 0 α and Σ 0 α and are contained in the sets Π 0 α+1 and Σ 0 α+1 , so that the difference sets live between the levels of the Borel hierarchy. All the classes above are pointclasses, that is, they are closed under inverse images by continuous functions.…”

“…• The set N CF of CF-normal numbers is Π 0 3 -complete [2]. • The set b≥2 N b of absolutely normal numbers is Π 0 3 -complete [7].…”