M{\"o}bius orthogonality in density for zero entropy dynamical systems

TL;DR

This paper proves that for certain zero entropy dynamical systems with countably many ergodic measures, the M{"o}bius function is orthogonal in density, extending M{"o}bius orthogonality to a broad class of systems.

Contribution

It establishes a density version of M{"o}bius orthogonality for zero entropy systems with countably many ergodic measures, a significant extension of prior results.

Findings

Existence of a subset of integers with logarithmic density one where orthogonality holds.

Uniform convergence of averages involving the M{"o}bius function and continuous functions.

Extension of M{"o}bius orthogonality to zero entropy systems with countably many ergodic measures.

Abstract

It is proved that whenever a zero entropy dynamical system $(X,T)$ has only countably many ergodic measures and $\mu$ stands for the arithmetic M{\"o}bius function, then there exists a subset $A$ of integers depending only on the system, of logarithmic density one, such that for each $f$ continuous on $X$, $\frac1N \sum_{n\leq N} f(T^nx)\mu(n) \to 0$ as $N\to\infty$, $N\in A$, uniformly in $x\in X$. In particular, the density version of M{\"o}bius orthogonality holds.