On uniformly approximable Sidon sets

Abstract

Let G be a compact abelian group and let T be the character group of G. Suppose £ is a subset of T. A trigonometric polynomial f on G is said to be an ^-polynomial if its Fourier transform / vanishes off E. The set E is said to be a Sidon set if there is a positive number B such that 2^xeb |/(X)| a-B||/||u for all E-polynomials /; here, ||/||„ = sup{ |/(x)| : xEG}. In this note we shall discuss a certain class of Sidon sets—the class of all approximate Sidon sets. A Sidon set E is said to be uniformly approximable if the characteristic function of E is in the uniform closure of the algebra of FourierStieltjes transforms of Radon measures on G. The principal result in this note is a theorem which gives a list of characterizations of uniformly approximable Sidon sets. These are somewhat akin to characterizations for Sidon sets given by Hewitt and Zuckerman [2, 2.1 and 8.5] and by Rudin ([7] and [6, 5.7]). Also, we shall propose several alternative characterizations and show (except in one case) that these would-be characterizations are false. Steckin [8, p. 394] established a sufficient condition for Sidon sets, which has been generalized successively by Hewitt and Zuckerman [2], Rudin [7], and Rider [5]. Evidently all known Sidon sets are finite unions of Sidon sets which satisfy Rider's condition and it will be shown that all of these finite unions are uniformly approximable Sidon sets. Ramirez [3] has studied subsets of T whose characteristic functions are in the uniform closure of the algebra of Fourier-Stieltjes transforms. Among the results presented in [3] are several theorems about uniformly approximable Sidon sets. We must establish certain conventions about notation. Throughout the paper G denotes an arbitrary compact abelian group and T denotes the character group of G. We denote the measure algebras of G and T by M{G) and M{T) respectively and the group algebra of Haar-integrable functions on G by Z-i(G). We denote by C{G) the algebra of all complex-valued continuous functions on G. If A QT, then A' is the complement of A, %A is the characteristic function of A in T, and C0{A) denotes the set of all complex-valued functions on A which vanish at infinity. For A C.DQT, we let C{D, A) be the set of