Remarks on Haar meager sets and Haar null sets in spaces of sequences

TL;DR

This paper explores the distinctions between Haar meager and Haar null sets in sequence spaces, providing constructions that separate these concepts and highlighting their differences in spaces like l_p, c_0, and c.

Contribution

It introduces methods to construct sets that are Haar meager but not Haar null, and vice versa, in classical sequence spaces, clarifying their relationship.

Findings

Haar meager sets can be non-Haar null in sequence spaces.

Haar null sets can be non-Haar meager in these spaces.

The paper provides explicit constructions demonstrating these differences.

Abstract

In the paper we will show how to construct a Haar meager set (consequently meager) which is not Haar null, and conversely, a meager Haar null set which is not Haar meager in spaces of sequences: $l_p$ with $p\geq1$, $c_0$ or $c$. It refers to the paper \cite{Darji}.