Entropy of convolutions on the circle

Given ergodic p-invariant measures {mu(i)} on the 1-torus T = R/Z, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution mu(1) *...* mu(n) converges to log p. We also prove a variant o f this result for joinings of full entropy on T-N. In conjunction with a me thod of Host, this yields the following. Denote sigma(q)(x) = qx (mod 1). T hen for every p-invariant ergodic mu with positive entropy, 1/N Sigma(n=0)( N-1) sigma(cn)mu converges weak* to Lebesgue measure as N --> infinity, und er a certain mild combinatorial condition on {c(k)}. (For instance, the con dition is satisfied if p = 10 and c(k) = 2(k) + 6(k) or c(k) = 2(2k).) 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 Hausdorff dimension of su m sets: For any sequence (Si) of p-invariant closed subsets of T, if Sigma dim(H)(S-i)/\ log dim(H)(S-i)\ = infinity, then dim(H)(S-1 +...+ S-n) --> 1 .