Normality of the Thue–Morse sequence along Piatetski-Shapiro sequences, II

Abstract

Abstract. We prove that the Thue-Morse sequence t along subsequences indexed by ⌊n c ⌋ is normal, where 1 < c < 3/2. That is, for c in this range and for each ω ∈ {0, 1} L , where L ≥ 1, the set of occurrences of ω as a factor (contiguous finite subsequence) of the sequence n → t ⌊n c ⌋ has asymptotic density 2 −L . This is an improvement over a recent result by the second author, which handles the case 1 < c < 4/3.In particular, this result shows that for 1 < c < 3/2 the sequence n → t ⌊n c ⌋ attains both of its values with asymptotic density 1/2, which improves on the bound c < 1.4 obtained by Mauduit and Rivat (who obtained this bound in the more general setting of q-multiplicative functions, however) and on the bound c ≤ 1.42 obtained by the second author.In the course of proving the main theorem, we show that 2/3 is an admissible level of distribution for the Thue-Morse sequence, that is, it satisfies a Bombieri-Vinogradov type theorem for each exponent η < 2/3. This improves on a result by Fouvry and Mauduit, who obtained the exponent 0.5924. Moreover, the underlying theorem implies that every finite word ω ∈ {0, 1} L is contained as an arithmetic subsequence of t.

The level of distribution of the Thue–Morse sequence

“…In our earlier paper [MS17] with Müllner, this theorem is proved via a Beatty sequence variant of Theorem 1.1. That theorem in turn is proved by arguments analogous to the arguments in the proof of Theorem 1.1 and reduces to the same estimate of the Gowers uniformity norm of Thue–Morse.…”

“…Müllner and the author [MS17] improved the exponent to , thereby establishing as an admissible level of distribution of the Thue–Morse sequence.…”

The sum-of-digits function on arithmetic progressions

“…Moreover, the first author was supported by the FWF project F5502-N26, which is a part of the Special Research Program "Quasi Monte Carlo methods: Theory and Applications"; the second author was supported by the project ANR-18-CE40-0018. This is a generalization of the statement that the Thue-Morse sequence t has full arithmetic complexity, meaning that every finite word ω ∈ {0, 1} L occurs as an arithmetic subsequence of t. This was first proved in [1] and also follows from Müllner and the first named author [8], and Konieczny [7]. Theorem 1.1 is not hard to prove for m = 1.…”

Kloosterman sums with twice-differentiable functions

“…‚ smooth, rough, and square-free numbers [1,2,6], ‚ almost-primes [7,9], ‚ additive problems [4,20,22], ‚ intersection with special sequences [3,5,8,13,19], and ‚ digital expansions [21,24,28]. For a broader picture of the scope of each topic, we refer the interested reader to the references within the cited items.…”