Towards a sharp converse of Wall’s theorem on arithmetic progressions

Abstract

Wall's theorem on arithmetic progressions says that if 0.a1a2a3 . . . is normal, then for any k, ℓ ∈ N, 0.a k a k+ℓ a k+2ℓ . . . is also normal. We examine a converse statement and show that if 0.an 1 an 2 an 3 . . . is normal for periodic increasing sequences n1 < n2 < n3 < . . . of asymptotic density arbitrarily close to 1, then 0.a1a2a3 . . . is normal. We show this is close to sharp in the sense that there are numbers 0.a1a2a3 . . . that are not normal, but for which 0.an 1 an 2 an 3 . . . is normal along a large collection of sequences whose density is bounded a little away from 1.

Finite-State Relative Dimension, Dimensions of AP Subsequences and a Finite-State van Lambalgen’s Theorem

“…Weiss and Kamae ([18] and [10]) showed that a number is normal if and only normality is preserved along every deterministic subsequence with positive asymptotic lower density. In a recent work, Vandehey [16] proved a nearly sharp converse to the Wall's theorem. Vandehey considers collections of subsequences such that for any ǫ, the collection contains a subsequence with asymptotic lower density greater than 1 − ǫ. Theorem 1.3 in [16] shows that preservation of normality along all subsequences in such a collection implies the normality of the original number.…”

“…In a recent work, Vandehey [16] proved a nearly sharp converse to the Wall's theorem. Vandehey considers collections of subsequences such that for any ǫ, the collection contains a subsequence with asymptotic lower density greater than 1 − ǫ. Theorem 1.3 in [16] shows that preservation of normality along all subsequences in such a collection implies the normality of the original number. Theorem 1.4 from the same paper shows that this converse to the Wall's theorem is close to being sharp.…”

“…Weiss [18], Kamae [10] and Vandehey [16] show that expanding the set of subsequences along which normality is investigated can yield interesting answers in the converse direction. Weiss and Kamae ([18] and [10]) showed that a number is normal if and only normality is preserved along every deterministic subsequence with positive asymptotic lower density.…”