Finding lower bounds on the growth and entropy of subshifts over countable groups

TL;DR

This paper presents a simpler and more effective lower bound on the growth and entropy of subshifts over countable groups, improving upon previous results and applying to various problems like aperiodicity and Kolmogorov complexity.

Contribution

It introduces a new, simpler lower bound method for subshift growth that outperforms previous approaches in certain settings.

Findings

Provides a simpler proof for lower bounds on subshift growth.

Offers improved bounds for applications in various subshift problems.

Demonstrates the effectiveness of the new bounds in multiple case studies.

Abstract

We give a lower bound on the growth of a subshift based on a simple condition on the set of forbidden patterns defining that subshift. Aubrun et Al. showed a similar result based on the Lov\'asz Local Lemma for subshift over any countable group and Bernshteyn extended their approach to deduce, amongst other things, some lower bound on the exponential growth of the subshift. However, our result has a simpler proof, is easier to use for applications, and provides better bounds on the applications from their articles (although it is not clear that our result is stronger in general). In the particular case of subshift over $\mathbb{Z}$ a similar but weaker condition given by Miller was known to imply nonemptiness of the associated shift. Pavlov used the same approach to provide a condition that implied exponential growth. We provide a version of our result for this particular setting and it is provably strictly stronger than the result of Pavlov and the result of Miller (and, in practice, leads to considerable improvement in the applications). We also apply our two results to a few different problems including strongly aperiodic subshifts, nonrepetitive subshifts, and Kolmogorov complexity of subshifts.