Uniform Szegő cocycles over strictly ergodic subshifts

We consider ergodic families of Verblunsky coefficients generated by minimal aperiodic subshifts. Simon conjectured that the associated probability measures on the unit circle have essential support of zero Lebesgue measure. We prove this statement for a large class of subshifts, namely those satisfying a condition originally introduced by Boshernitzan. This is accomplished by relating the essential support to uniform convergence properties of the corresponding Szegő cocycles.