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