On uniformity of q‐multiplicative sequences

Fan, Aihua; Konieczny, Jakub

doi:10.1112/blms.12245

Cited by 8 publications

(7 citation statements)

References 52 publications

(113 reference statements)

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“…For a similar result for a much simpler class of q-multiplicative sequences, see [FK19]. Examples of highly Gowers uniform sequences of number-theoretic origin in finite fields of prime order were found in [FKM13]; see also [Liu11] and [NR09] where Gowers uniformity of certain sequences is derived from much stronger discorrelation estimates.…”

Section: Introductionmentioning

confidence: 82%

Gowers norms for automatic sequences

Byszewski¹,

Konieczny²,

Müllner³

2020

Preprint

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We show that any automatic sequence can be separated into a structured part and a Gowers uniform part in a way that is considerably more efficient than guaranteed by the Arithmetic Regularity Lemma. For sequences produced by strongly connected and prolongable automata, the structured part is rationally almost periodic, while for general sequences the description is marginally more complicated. In particular, we show that all automatic sequences orthogonal to periodic sequences are Gowers uniform. As an application, we obtain for any l ≥ 2 and any automatic set A ⊂ N0 lower bounds on the number of l-term arithmetic progressions -contained in A -with a given difference. The analogous result is false for general subsets of N0 and progressions of length ≥ 5.

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Section: Introductionmentioning

confidence: 82%

Gowers norms for automatic sequences

Byszewski¹,

Konieczny²,

Müllner³

2020

Preprint

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“…The analysis of q-additive functions and their distribution (as well as their multiplicative counterpart) have attained a lot of attention during the last few decades, see for example [3,6,7,9,10,11,18,21,22,23,24,25,26,30].…”

Section: Introductionmentioning

confidence: 99%

Effective Erdős–Wintner theorems for digital expansions

Drmota

Verwee

2021

Journal of Number Theory

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In 1972 Delange [8] observed in analogy of the classical Erdős-Wintner theorem that q-additive functions f (n) has a distribution function if and only if the two series f (dq j ), f (dq j ) 2 converge. The purpose of this paper is to provide quantitative versions of this theorem as well as generalizations to other kinds of digital expansions. In addition to the q-ary and Cantor case we focus on the Zeckendorf expansion that is based on the Fibonacci sequence, where we provide a sufficient and necessary condition for the existence of a distribution function, namely that the two series f (F j ), f (F j ) 2 converge (previously only a sufficient condition was known [1]).

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“…Green and Tao [GT12, Theorem 1.1] proved that nilsystems satisfy polynomial Sarnak conjecture. Using Green and Tao's result and adopting the proof that the Möbius function is a good weight for the classical pointwise ergodic theorem [AKPLDLR14, Proposition 3.1] (see also [FK19, Theorem C]), Eisner [Eis18, Theorem 2.2] proved that (3) holds for f ∈ L q (ν) and for ν-a.e. x where q > 1 and ν is any T -invariant measure.…”

Section: Introductionmentioning

confidence: 99%