THE CLASS L(LOG L)(ALPHA) AND SOME LACUNARY SETS

A well-known result of Zygmund states that if f is an element of L (lo g(+) L)(1/2) on the circle group T and E is a Hadamard set of integers , then f\(E) is an element of l(2) (E). In this paper we investigate s imilar results for the classes B-alpha = L (log(+) L)(alpha), alpha > 0 on an arbitrary infinite compact abelian group G and Sidon subsets E of the dual Gamma. These results are obtained as special cases of mor e general results concerning a new class of lacunary sets S-alpha,S-be ta, 0 < alpha less than or equal to beta, where a subset E of Gamma is an S-alpha,S-beta set if B-alpha\(E) subset of or equal to l(2 beta/a lpha) (E). We also prove partial results on the distinctness of the S- alpha,S-beta sets in the index beta.