A generalization of the density zero ideal

TL;DR

This paper introduces a generalized ideal based on a sequence of finite sets and explores its properties, showing it is an analytic P-ideal but not F_sigma, and examines the implications for sequence convergence in functional spaces.

Contribution

It generalizes the classical density zero ideal using arbitrary finite sets and analyzes its topological and convergence properties.

Findings

The ideal is an analytic P-ideal.

It is not an F_sigma set.

The set of sequences converging under this ideal is not complemented in ℓ_infinity.

Abstract

Let $\mathscr{F}=(F_n)$ be a sequence of nonempty finite subsets of $\omega$ such that $\lim_n |F_n|=\infty$ and define the ideal $$\mathcal{I}(\mathscr{F}):=\left\{A\subseteq \omega: |A\cap F_n|/|F_n|\to 0~\mbox{as}~n\to \infty \right\}.$$ The case $F_n=\{1,\ldots,n\}$ corresponds to the classical case of density zero ideal. We show that $\mathcal{I}(\mathscr{F})$ is an analytic P-ideal but not $F_{\sigma}$. As a consequence, we show that the set of real bounded sequences which are $\mathcal{I}(\mathscr{F})$-convergent to $0$ is not complemented in $\ell_\infty$.