Entropy of Convolutions on the Circle

Given ergodic p-invariant measures f i g on the 1-torus T = R=Z, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution 1 n converges to log p. We also prove a variant of this result for joinings of full entropy on T N. In conjunction with a method of Host, this yields the following. Denote q (x) = qx (mod 1). Then for every p-invariant ergodic with positive entropy, 1 N P N?1 n=0 cn converges weak to Lebesgue measure as N ?! 1, under a certain mild combinatorial condition on fc k g. (For instance, the condition is satissed if p = 10 and c k = 2 k + 6 k or c k = 2 2 k .) This extends a result of Johnson and Rudolph, who considered the sequence c k = q k when p and q are multiplicatively independent. We also obtain the following corollary concerning Hausdorr dimension of sum sets: For any sequence fS i g of p-invariant closed subsets of T, if P dim H (S i)=j log dim H (S i)j = 1, then dim H (S 1 + + S n) ?! 1.