Minimal zero entropy subshifts can be unrestricted along any sparse set

TL;DR

The paper proves that minimal zero-entropy subshifts can exhibit arbitrary behavior on sparse sets, providing counterexamples to the Polynomial Sarnak Conjecture and demonstrating limitations of minimality and zero entropy assumptions.

Contribution

It offers a streamlined proof of the existence of minimal zero-entropy subshifts with arbitrary behavior on zero Banach density sets, extending previous results.

Findings

Constructs minimal zero-entropy subshifts with arbitrary restrictions on sparse sets.

Provides counterexamples to the Polynomial Sarnak Conjecture.

Shows limitations of minimality and zero entropy assumptions.

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 $(X, \sigma)$ 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, Lema\'{n}czyk, and Radziwi\l\l and of Lian and Shi, and shows that no similar result can hold under only the assumptions of minimality and zero entropy.