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In this paper, we show that for any sequence ${\bf a}=(a_n)_{n\in \Z}\in\{1,\ldots,k\}^\mathbb{Z}$ and any $\epsilon>0$, there exists a Toeplitzsequence ${\bf b}=(b_n)_{n\in \Z}\in \{1,\ldots,k\}^\mathbb{Z}$ such that theentropy $h({\bf b})\leq 2 h({\bf a})$ and$\lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^N|a_n-b_n|<\epsilon$. As anapplication of this result, we reduce Sarnak Conjecture to Toeplitz systems,that is, if the M\"{o}bius function is disjoint from any Toeplitz sequence withzero entropy, then the Sarnak conjecture holds.

Reducing the Sarnak Conjecture to Toeplitz systems