  Rotations on the circle by irrational numbers give rise to uniquely ergodic
Sturm dynamical systems. We show that rotations by badly approximable
irrationals have the property of fast ergodicity. It was shown recently that
any Sturmian ergodic measure is the unique ground state of a non-frustrated
Hamiltonian of one-dimensional classical lattice-gas model. We use the fast
ergodicity property to show that for slowly decaying interactions,
$1/r^{\alpha}$ with $1 < \alpha < 3/2$, non-periodic Sturmian ground states are
stable with respect to periodic configurations consisting of Sturmian words.
