Almost full entropy subshifts uncorrelated to the M\"obius function

TL;DR

This paper constructs high-entropy subshifts over finite symbols that are uncorrelated with sequences like the M"obius function, providing explicit examples supporting Sarnak's conjecture and clarifying their algebraic independence.

Contribution

It demonstrates the existence of high-entropy subshifts uncorrelated with the M"obius function, filling a gap in the literature and showing this is unrelated to algebraic properties.

Findings

Existence of subshifts with entropy close to log N uncorrelated with given sequences.

Construction of examples uncorrelated with the Mf6bius function.

Uncorrelated subshifts satisfy Sarnak's conjecture.

Abstract

We show that if $y=(y_n)_{n\ge 1}$ is a bounded sequence with zero average along every infinite arithmetic progression then for every $N\ge 2$ there exist (unilateral or bilateral) subshifts $\Sigma$ over $N$ symbols, with entropy arbitrarily close to $\log N$, uncorrelated to $y$. In particular, for $y=\mu$ being the M\"obius function, we get that there exist subshifts as above which satisfy the assertion of Sarnak's conjecture. The existence of positive entropy systems uncorrelated to the M\"obius function is claimed in Sarnak's survey \cite{sarnak} (and attributed to Bourgain), however, to our knowledge no examples have ever been published. We fill in this gap and by the way we show that this has nothing to do with more advanced algebraic properties (for instance multiplicativity) of the considered sequence.