Analytic and arithmetic properties of thin sets

John J. F. Fournier and Louis Pigno

Vol. 105 (1983), No. 1, 115–141

DOI: 10.2140/pjm.1983.105.115

Abstract

Let Z⁺ 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⁻∪ E is a set of continuity. For instance, if E is a Λ(1) set, a p-Sidon set (for some p < 2), or a UC-set, then E cannot contain parallelepipeds of arbitrarily large dimension, and it then follows that Z⁻∪ E 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.

Mathematical Subject Classification 2000

Primary: 43A46

Milestones

Received: 25 February 1981

Revised: 16 February 1982

Published: 1 March 1983