Minimal zero entropy subshifts can be unrestricted along any sparse set

Abstract

We present a streamlined proof of a result essentially present in previous work of the author, namely that for every set $S = \{s_1, s_2, \ldots\} \subset \mathbb{N}$ of zero Banach density and finite set $A$, there exists a minimal zero-entropy subshift $(X, σ)$ so that for every sequence $u \in A^\mathbb{Z}$, there is $x_u \in X$ with $x_u(s_n) = u(n)$ for all $n \in \mathbb{N}$. Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the Polynomial Sarnak Conjecture which are significantly more general than some recently provided in word of Kanigowski, Lemańczyk, and Radziwiłłand of Lian and Shi, and shows that no similar result can hold under only the assumptions of minimality and zero entropy.