Building Large Free Subshifts Using the Local Lemma

TL;DR

This paper demonstrates that large free subshifts with significant Hausdorff dimension and entropy exist in countably infinite groups, using the Lovász Local Lemma to strengthen previous results and answer open questions.

Contribution

It extends prior work by constructing large free subshifts with specified Hausdorff dimension and entropy, employing probabilistic combinatorics techniques.

Findings

Existence of free subshifts with arbitrary Hausdorff dimension less than log2(k)

Existence of free subshifts with entropy at least h for sofic groups

A general lower bound on the size of free subshifts within given subshifts

Abstract

Gao, Jackson, and Seward proved that every countably infinite group $\Gamma$ admits a nonempty free subshift $X \subseteq 2^\Gamma$. Here we strengthen this result by showing that free subshifts can be "large" in various senses. Specifically, we prove that for any $k \geqslant 2$ and $h < \log_2 k$, there exists a free subshift $X \subseteq k^\Gamma$ of Hausdorff dimension and, if $\Gamma$ is sofic, entropy at least $h$, answering two questions attributed by Gao, Jackson, and Seward to Juan Souto. Furthermore, we establish a general lower bound on the largest "size" of a free subshift $X'$ contained inside a given subshift $X$. A central role in our arguments is played by the Lov\'{a}sz Local Lemma, an important tool in probabilistic combinatorics, whose relevance to the problem of finding free subshifts was first recognized by Aubrun, Barbieri, and Thomass\'{e}.