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arXiv · 2202.09491

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

Amir Algom·Zhiren Wang
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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.

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Amir Algom, Zhiren Wang. 2022-02-19. Arbitrarily slow decay in the M\"{o}bius disjointness conjecture. https://arxiv.org/abs/2202.09491

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