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Polynomial mean complexity and logarithmic Sarnak conjecture

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Ergodic theory

Published online by Cambridge University Press:  27 April 2023

WEN HUANG ,
LEIYE XU  and
XIANGDONG YE
WEN HUANG
Affiliation:
CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China (e-mail: wenh@mail.ustc.edu.cn, yexd@ustc.edu.cn)
LEIYE XU*
Affiliation:
CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China (e-mail: wenh@mail.ustc.edu.cn, yexd@ustc.edu.cn)
XIANGDONG YE
Affiliation:
CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China (e-mail: wenh@mail.ustc.edu.cn, yexd@ustc.edu.cn)
*
e-mail: leoasa@mail.ustc.edu.cn
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Abstract

In this paper, we reduce the logarithmic Sarnak conjecture to the $\{0,1\}$-symbolic systems with polynomial mean complexity. By showing that the logarithmic Sarnak conjecture holds for any topologically dynamical system with sublinear complexity, we provide a variant of the $1$-Fourier uniformity conjecture, where the frequencies are restricted to any subset of $[0,1]$ with packing dimension less than one.

Keywords

Möbius functiontopological dynamicspolynomial complexitypacking dimension

MSC classification

Primary: 37A35: Entropy and other invariants, isomorphism, classification 11K31: Special sequences
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 3 , March 2024 , pp. 769 - 798
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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