  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 \muon 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 \sig_q(x) = qx\pmod{1}. Then for every
p-invariant ergodic \mu with positive entropy,
\frac{1}{N}\sum_{n=0}^{N-1}\sig_{c_n}\mu converges weak^* to Lebesgue measure
as N \goesto \infty, 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^{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 Hausdorff dimension of sum
sets: For any sequence {S_i} of p-invariant closed subsets of T, if \sum
\dim_H(S_i) / |\log\dim_H(S_i)| = \infty, then \dim_H(S_1 + \cdots + S_n)
\goesto 1.
