Interpolation and equicontinuity sets in topological groups and spaces of continuous functions

Abstract

Index 105Bibliography 109 IntroductionClassical Hadamard sets (also called lacunary sets) have been studied since the beginning of the past century. Definition.A subset E = {n j } ∞ j=1 of N, with n 1 < n 2 < . . . is said to be a Hadamard set if there exists some q > 1 (called Hadamard ratio) such that n j+1 /n j ≥ q for all 1 ≤ j ≤ ∞. c j z n j has a radius of convergence equal to 1. Then f cannot be analytically continued across any portion of the arc |z| = 1.In the 1950's, some of these properties were more closely studied and finally became definitions. Since these properties were more functional analytic in nature, whereas the original lacunary property was arithmetic, it was no longer necessary to restrict attention to sets of integers.Therefore, lacunary sets have also been studied in the dual set of compact topological groups and in more general settings. Kahane [62] first used the term Sidon set in 1957 and the modern point of view of the notion appeared in Rudin's Book [86] in 1962. Let G = T, a Hadamard set E ⊆ G = Z is a special type of Sidon set.Given a compact abelian group G, one of the several characterisations of a Sidon set (see [66], for instance), stated that a subset E of G is a Sidon set if every bounded function on E is the restriction of the Fourier transform of a measure on G.Since the dual of a compact group is a discrete group and the Sidon sets are situated in the dual set, we can study the Sidon sets as subsets of discrete groups. In this sense, Picardello [76] extended in 1973 the usual definition of Sidon set in a group G to the discrete non-abelian case. 2 IntroductionA special kind of Sidon set is the I 0 sets. The concept of I 0 set in locally compact abelian groups was introduced by Hartman and Ryll-Nardzewski [48] in 1964, who considered the weak topology associated to a locally compact abelian (LCA, for short) group and introduced the notion of interpolation set or I 0 set. They defined that a subset A of a LCA group G is an I 0 set if every bounded function on A is the restriction of an almost periodic function on G (here, it is said that a complex-valued function f defined on G is almost periodic when it is the restriction of a continuous function defined on bG, the Bohr compactification of G).Therefore, an I 0 set is a subset A of G such that any bounded map on A can be interpolated by a continuous function on bG. As a consecuence, if A is a countably infinite I 0 set, then A bG is canonically homeomorphic to βω, the Stone-Čech compactification of ω. The main result given by Hartman and Ryll-Nardzewski is the following:For the particular case of discrete abelian groups, van Douwen achieved a remarkable progress by proving the existence of I 0 sets in very general situations. His main result can be formulated in the following way:Theorem. ([94, Th. 1.1.3]) Let G be a discrete Abelian group and let A be an infinite subset of G. Then, there is a subset B of A with |B| = |A| such that B is an I 0 set.In fact, van Douwen extended his result to the real line but left unresolv...