Arbitrarily slow decay in the logarithmically averaged Sarnak conjecture

TL;DR

This paper constructs examples demonstrating that Tao's logarithmic average version of Sarnak's conjecture can decay arbitrarily slowly, yet still satisfy the conjecture, highlighting nuanced behavior in the conjecture's decay rate.

Contribution

The paper provides explicit examples showing the decay in Tao's logarithmic Sarnak conjecture can be arbitrarily slow while still satisfying the conjecture.

Findings

Decay rate can be arbitrarily slow

Examples satisfy the conjecture despite slow decay

Highlights subtlety in the conjecture's decay behavior

Abstract

In 2017 Tao proposed a variant Sarnak's M\"{o}bius disjointness conjecture with logarithmic averaging: For any zero entropy dynamical system $(X,T)$, $\frac{1}{\log N} \sum_{n=1} ^N \frac{f(T^n x) \mu (n)}{n}= o(1)$ for every $f\in \mathcal{C}(X)$ and every $x\in X$. We construct examples showing that this $o(1)$ can go to zero arbitrarily slowly. Nonetheless, all of our examples satisfy the conjecture.