p-Sidon sets

Abstract

Let G be a compact Abelian group with character group X. Bożejko and Pytlik [Colloq. Math. 25 (1972), 117-124] introduced and studied several special types of lacunary subsets of X. This paper is based upon a hitherto unpublished detailed study of those types that most resemble Sidon sets, which the present authors had independently introduced and studied under the name of p-Sidon sets. Some, but not all, aspects of the theory of Sidon (= 1-Sidon) sets carry over to the more general setting. In Section 1 some properties of sets equivalent to p-Sidonicity are given. Section 2 contains several useful consequences of p-Sidonicity; see Theorems 2.1 and 2.4 and Corollaries 2.6 and 2.7. In Section 3, it is shown that certain Λq sets also satisfy some of the consequences listed in Section 2. Nevertheless, Λq sets need not be p-Sidon sets; see Theorem 3.1. Examples of ( 4 3)-Sidon sets that are not Sidon sets are given in Section 5. The proof that these sets are ( 4 3)-Sidon sets requires a brief study of 4-norms in Varopoulos algebras; see Section 4. In Section 6, some special results for the circle group are deduced. Many of these results appear to be new even for p = 1. © 1974.