Arbitrarily slow decay in the Möbius disjointness conjecture

2022/02/19 by Amir Algom, Zhiren Wang, Algom, Amir +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Benford’s Law and Fraud Detection #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2202.09491

openalex publication_date 2022/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Sarnak's Möbius disjointness conjecture asserts that for any zero entropy dynamical system (X,T), (1)/(N) ∑_n=1 ^N f(Tⁿ x) μ(n)= o(1) for every f∈ C(X) and every x∈ X. We construct examples showing that this o(1) can go to zero arbitrarily slowly. In fact, our methods yield a more general result, where in lieu of μ(n) one can put any bounded sequence such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero.

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