Entropy of Convolutions on the Circle

Abstract

Given ergodic p-invariant measures f„ig on the 1-torus = = ,w e give a sharp condition on their entropies, guaranteeing that the entropy of the convolution „1⁄¢¢¢⁄„n converges to logp. We also prove a variant of this result for joinings of full entropy on . In conjunction with a method of Host, this yields the following. Denote aeq(x )= qx (mod 1). Then for every p-invariant ergodic „ with positive entropy, 1 N P Ni1 n=0 aecn „ converges weak ⁄ to Lebesgue measure as N i! 1 , under a certain mild combinatorial condition onfckg. (For instance, the condition is satisfled if p = 10 and ck =2 k +6 k or ck =2 2 k .) This extends a result of Johnson and Rudolph, who considered the sequence ck = q k when p and q are multiplicatively independent. We also obtain the following corollary concerning Hausdorfi dimension of sum sets: For any sequence fSig of p-invariant closed subsets of ,i f P dimH(Si)=j log dimH(Si)j =1, then dimH(S1 +¢¢¢+Sn)i! 1.