Minimal zero entropy subshifts can be unrestricted along any sparse set
Ronnie Pavlov
TL;DR
The paper addresses whether minimal zero-entropy systems can enforce correlation with arithmetic functions along sparse sets. It provides a streamlined, self-contained block-concatenation construction that yields a minimal zero-entropy subshift $X\subset A^{\\mathbb{Z}}$ for which, for every $u\in A^{\\mathbb{N}}$, there exists $x_u\in X$ with $x_u(s_n)=u(n)$ on a zero-density set $S$ (i.e., $d^*(S)=0$). This leads to strong corollaries refuting the Polynomial Sarnak Conjecture in a broad setting and shows that minimality plus zero entropy do not guarantee any form of regular averaging along sparse sets. The construction also demonstrates the boundary case: if $d^*(S)>0$, then the resulting subshift must have positive entropy, highlighting the necessity of zero density for the main result.
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.