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

Abstract

We provide a lower bound on the growth of a subshift based on a simple condition on the set of forbidden patterns defining that subshift. Aubrun, Barbieri, and Thomassé showed a similar result based on the Lovász local lemma for subshifts over any countable group, and Bernshteyn extended their approach to deduce some lower bound on the exponential growth of the subshift. 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 strictly stronger in general). In the particular case of subshifts over $\mathbb{Z}$, Miller gave a similar but weaker condition that implied the non-emptiness 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. In practice, it 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.