Reducing the Sarnak Conjecture to Toeplitz systems

Abstract

In this paper, we show that for any sequence $\mathbf{a}=(a_n{)}_{n\in \mathbb{Z}}\in \{1,\dots ,k{\}}^{\mathbb{Z}}$ and any $\epsilon >0$, there exists a Toeplitz sequence $\mathbf{b}=(b_n{)}_{n\in \mathbb{Z}}\in \{1,\dots ,k{\}}^{\mathbb{Z}}$ such that the entropy $h(\mathbf{b})\le 2h(\mathbf{a})$ and ${\lim }_{N\to \infty }\frac{1}{2N+1}{\sum }_{n=-N}^N|a_n-b_n|<\epsilon$. As an application of this result, we reduce Sarnak Conjecture to Toeplitz systems, that is, if the Möbius function is disjoint from any Toeplitz sequence with zero entropy, then the Sarnak conjecture holds.

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