Analytic and arithmetic properties of thin sets

Abstract

+ and Z~ denote the sets of positive and negative integers respectively. We study relations between various thinness conditions on subsets E of Z + , with particular emphasis on those conditions that imply Z~U is a set of continuity. For instance, if E is a (l) set, a /?-Sidon set (for some p < 2), or a /C-set, then E cannot contain parallelepipeds of arbitrarily large dimension, and it then follows that Z~UE is a set of continuity; on the other hand there is a set E that is Rosenthal, strong Riesz, and Rajchman, which is not a set of continuity.