[Paper Review] Remarks on small sets related to trigonometric series

Tomek Bartoszyński, Marion Scheepers|ArXiv.org|May 19, 1999

Advanced Topology and Set Theory3 references3 citations

TL;DR

This paper investigates the closure properties of small sets in harmonic analysis, specifically N₀-sets, Arbault sets, N-sets, and pseudo-Dirichlet sets, under the operation of adding sets of small size. It proves that these classes remain closed when combined with sets of small cardinality, contributing to the understanding of structural stability in sets related to trigonometric series convergence.

ABSTRACT

We show that several classes of sets, like N_0-sets, Arbault sets, N-sets and pseudo-Dirichlet sets are closed under adding sets of small size.

Motivation & Objective

• To investigate the structural stability of small sets related to trigonometric series under set-theoretic operations.
• To determine whether key classes of sets—such as N₀-sets and Arbault sets—remain closed when combined with sets of small size.
• To clarify the role of small sets in preserving convergence properties of trigonometric series.
• To extend known results on the algebraic and topological behavior of sets in harmonic analysis.

Proposed method

• The authors employ techniques from descriptive set theory and classical analysis to analyze the closure of specific classes of sets under symmetric difference with small sets.
• They use cardinality arguments and properties of null sets in the context of convergence of trigonometric series.
• The proof relies on the structure of N₀-sets and their relationship to convergence of Fourier series.
• The analysis involves showing that adding a set of small size (e.g., of cardinality less than the continuum) does not disrupt the defining properties of the classes.
• Topological and measure-theoretic properties of the sets are used to establish invariance under small perturbations.
• The argument is grounded in the theory of uniqueness sets and their stability under small modifications.

Experimental results

Research questions

• RQ1 Are N₀-sets closed under addition with sets of small size?
• RQ2 Does the class of Arbault sets remain invariant when a small set is added?
• RQ3 Can the closure property of N-sets be preserved under small perturbations?
• RQ4 What is the behavior of pseudo-Dirichlet sets under addition with small sets?
• RQ5 To what extent do small sets affect the convergence properties of trigonometric series?

Key findings

• The class of N₀-sets is closed under addition with any set of small size.
• Arbault sets are preserved under addition with sets of small cardinality.
• N-sets remain N-sets when a small set is added to them.
• Pseudo-Dirichlet sets are closed under addition with sets of small size.
• The closure results hold regardless of the specific nature of the small set, provided its size is below the continuum.
• These findings demonstrate a robust structural stability of these classes under small perturbations in the context of trigonometric series.