Arbitrarily slow decay in the M\"{o}bius disjointness conjecture

TL;DR

This paper demonstrates that the convergence in Sarnak's Möbius disjointness conjecture can be arbitrarily slow, providing examples that challenge the expected rate of decay in zero entropy dynamical systems.

Contribution

It constructs examples showing the decay in Möbius disjointness can be made arbitrarily slow, extending to general bounded sequences with non-vanishing Cesàro means.

Findings

Decay rate in Möbius disjointness can be arbitrarily slow.

Constructs examples for zero entropy systems with slow convergence.

Generalizes to bounded sequences with non-zero Cesàro means.

Abstract

Sarnak's M\"{o}bius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$, $\frac{1}{N} \sum_{n=1} ^N f(T^n x) \mu (n)= o(1)$ 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. In fact, our methods yield a more general result, where in lieu of $\mu(n)$ one can put any bounded sequence such that the Ces\`aro mean of the corresponding sequence of absolute values does not tend to zero.