THIN SETS OF HARMONIC ANALYSIS AND INFINITE COMBINATORICS

Abstract

This is a survey paper on some classical trigonometric families of thin sets (Dirichlet sets, weak Dirichlet sets, N-sets, N₀-sets, A-sets, U-sets, and two recently introduced families of B-sets and of B₀-sets), the relationships between them, and basic closure properties of these families, presented as complete answers to ten questions. However, a large part of the paper is devoted to presentation of new results. In addition, we tried to give an overview of the best known estimates for cardinal characteristics for these families and for the families of particularly "permitted" sets, using small uncountable cardinals recently studied in infinite combinatorics. Almost all results are accompanied by brief notes on the investigations preceding them. Finally, we study properties of families of thin sets related to the Rademacher and Walsh orthogonal systems of functions. Some of these families are studied for the first time.