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The level of distribution of the Thue–Morse sequence

Part of: Sequences and sets Elementary number theory Multiplicative number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  25 January 2021

Lukas Spiegelhofer
Lukas Spiegelhofer*
Affiliation:
Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstrasse 8–10, 1040Vienna, Austrialukas.spiegelhofer@unileoben.ac.at Department of Mathematics and Information Technology, Montanuniversität Leoben, Franz-Josef-Straße 18, 8700Leoben, Austria
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Abstract

The level of distribution of a complex-valued sequence $b$ measures the quality of distribution of $b$ along sparse arithmetic progressions $nd+a$. We prove that the Thue–Morse sequence has level of distribution $1$, which is essentially best possible. More precisely, this sequence gives one of the first nontrivial examples of a sequence satisfying a Bombieri–Vinogradov-type theorem for each exponent $\theta <1$. This result improves on the level of distribution $2/3$ obtained by Müllner and the author. As an application of our method, we show that the subsequence of the Thue–Morse sequence indexed by $\lfloor n^c\rfloor$, where $1 < c < 2$, is simply normal. This result improves on the range $1 < c < 3/2$ obtained by Müllner and the author and closes the gap that appeared when Mauduit and Rivat proved (in particular) that the Thue–Morse sequence along the squares is simply normal.

Keywords

Thue–Morse sequencelevel of distributionBombieri–Vinogradov theoremElliott–Halberstam conjecturearithmetic progressionPiatetski-Shapiro sequencenormal sequenceGel'fond problem

MSC classification

Primary: 11N69: Distribution of integers in special residue classes 11B25: Arithmetic progressions 11B85: Automata sequences
Secondary: 11A63: Radix representation; digital problems 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11B83: Special sequences and polynomials
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 12 , December 2020 , pp. 2560 - 2587
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Copyright
© The Author(s) 2021

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Footnotes

The author acknowledges support by the Austrian Science Fund (FWF), Project F5502-N26, which is a part of the Special Research Program ‘Quasi Monte Carlo Methods: Theory and Applications’. The author also wishes to acknowledge support by the project MuDeRa, which is a joint project between the FWF (I-1751-N26) and the ANR (Agence Nationale de la Recherche, France, ANR-14-CE34-0009).

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