Realization of aperiodic subshifts and uniform densities in groups

A theorem of Gao, Jackson and Seward, originally conjectured to be false by Glasner and Uspenskij, asserts that every countable group admits a 2-coloring. A direct consequence of this result is that every countable group has a strongly aperiodic subshift on the alphabet 0; 1. In this article, we use Lovász local lemma to first give a new simple proof of said theorem, and second to prove the existence of a G-effectively closed strongly aperiodic subshift for any finitely generated group G. We also study the problem of constructing subshifts which generalize a property of Sturmian sequences to finitely generated groups. More precisely, a subshift over the alphabet 0; 1 has uniform density ? 2 OE0; 1. If for every configuration the density of 1's in any increasing sequence of balls converges to ?. We show a slightly more general result which implies that these subshifts always exist in the case of groups of subexponential growth.