Almost everywhere convergence of convolution powers on compact Abelian groups

It is well-known that a probability measure μ on the circle T satisfies μn * f f dmp → 0 for every f ∈ Lp, every (some) p ∈ [1,∞), if and only if | μ(n)| < 1 for every non-zero n ∈ Z (μ is strictly aperiodic). In this paper we study the a.e. convergence of μn * f for every f ∈ Lp whenever p >1. We prove a necessary and sufficient condition, in terms of the Fourier- Stieltjes coefficients of μ, for the strong sweeping out property (existence of a Borel set B with lim supμn * 1B = 1 a.e. and lim infμn * 1B = 0 a.e.). The results are extended to general compact Abelian groups G with Haar measure m, and as a corollary we obtain the dichotomy: for μ strictly aperiodic, either μn * f → f dm a.e. for every p >1 and every f ∈ Lp(G,m), or μ has the strong sweeping out property.