  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 Lov\'asz 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 $\alpha \in [0,1]$ if for every
configuration the density of $1$'s in any increasing sequence of balls
converges to $\alpha$. We show a slightly more general result which implies
that these subshifts always exist in the case of groups of subexponential
growth.
