The class l(log l)α and some lacunary sets

A well-known result of Zygmund states that if ([formula ommitted]) on the circle group T and £ is a Hadamard set of integers, then ([formula ommitted]) In this paper we investigate similar results for the classes ([formula ommitted]) on an arbitrary infinite compact abelian group G and Sidon subsets E of the dual T. These results are obtained as special cases of more general results concerning a new class of lacunary sets([formula ommitted]) where a subset E of ([formula ommitted]). We also prove partial results on the distinctness of the ([formula ommitted]) sets in the index p.