Tauberian theorems for ordinary convergence

TL;DR

This paper characterizes real sequence convergence using regular matrices and specific ideals, linking subsequence behavior to classical convergence and extending Steinhaus' theorem with new methods.

Contribution

It introduces a novel characterization of convergence via $ ext{I}$-convergence and regular matrices, employing recent $ ext{I}$-Baire class and filter game results.

Findings

Sequences are convergent iff certain subsequences are $ ext{I}$-convergent for a large set of subsequences.

Recovers classical results for finite and density zero ideals.

Provides a stronger version of Steinhaus' theorem with explicit non-statistically convergent sequences.

Abstract

We show that a real sequence $x$ is convergent if and only if there exist a regular matrix $A$ and an $F_{\sigma\delta}$-ideal $\mathcal{I}$ on $\mathbf{N}$ such that the set of subsequences $y$ of $x$ for which $Ay$ is $\mathcal{I}$-convergent is of the second Baire category. This includes the cases where $\mathcal{I}$ is the ideal of asymptotic density zero sets, the ideal of Banach density zero sets, and the ideal of finite sets. The latter recovers an old result given by Keogh and Petersen in [J. London Math. Soc. \textbf{33} (1958), 121--123]. Our proofs are of a different nature and rely on recent results in the context of $\mathcal{I}$-Baire classes and filter games. As application, we obtain a stronger version of the classical Steinhaus' theorem: for each regular matrix $A$, there exists a $\{0,1\}$-valued sequence $x$ such that $Ax$ is not statistically convergent.