Reducing the Sarnak Conjecture to Toeplitz systems

2019/08/20 by Wen Huang, Huang, Wen, Zhengxing Lian +5
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.DS

paper · pdf · doi:10.48550/arxiv.1908.07554

openalex publication_date 2019/08/20 · arxiv created 2019/08/22 · arxiv updated 2019/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we show that for any sequence \bf a=(a_n)_{n∈ \Z}∈ \1,…,k\^ℤ and any ε>0, there exists a Toeplitz sequence \bf b=(b_n)_{n∈ \Z}∈ \1,…,k\^ℤ such that the entropy h(\bf b)≤ 2 h(\bf a) and lim_N→∞(1)/(2N+1)∑_n=-N^N|a_n-b_n|<ε. As an application of this result, we reduce Sarnak Conjecture to Toeplitz systems, that is, if the Möbius function is disjoint from any Toeplitz sequence with zero entropy, then the Sarnak conjecture holds.

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