Polynomial mean complexity and Logarithmic Sarnak conjecture

TL;DR

This paper links the logarithmic Sarnak conjecture to systems with polynomial mean complexity, proving it for systems with sublinear complexity and proposing a variant of the Fourier uniformity conjecture with restricted frequencies.

Contribution

It reduces the logarithmic Sarnak conjecture to polynomial mean complexity systems and proves it for systems with sublinear complexity, introducing a new Fourier uniformity variant.

Findings

Logarithmic Sarnak conjecture holds for systems with sublinear complexity

Reduction of the conjecture to polynomial mean complexity systems

A variant of the Fourier uniformity conjecture with restricted frequencies

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.