Entropy of convolutions on the circle

Given ergodic p-invariant measures {μ_i} on the 1-torus double-struck T sign = double-struck R sign/double-struck Z sign, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution μ₁ * ⋯ * μ_n converges to log p. We also prove a variant of this result for joinings of full entropy on double-struck T sign^{double-struck N sign}. In conjunction with a method of Host, this yields the following. Denote σ_q(cursive Greek chi) = qcursive Greek chi (mod 1). Then for every p-invariant ergodic μ with positive entropy, 1/N Σ^N-1_n=0 σ_cnμ converges weak* to Lebesgue measure as N → ∞, under a certain mild combinatorial condition on {c_k}. (For instance, the condition 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 sum sets: For any sequence {S_i} of p-invariant closed subsets of double-struck T sign, if Σ dim_H(S_i)/| log dim_H (S_i)| = ∞, then dim_H(S₁ + ⋯ + S_n) → 1.