Density cardinals

TL;DR

This paper investigates the density number related to permutations of infinite sets, establishing its equality with the least size of a non-meager set of reals, and explores variants and related cardinal invariants.

Contribution

It proves the density number equals the non-meager set size, relates it to existing invariants, and analyzes variants with bounds and consistency results.

Findings

Density number equals the least size of a non-meager set of reals.

A modified rearrangement number also equals the non-meager set size.

Provides bounds and consistency results for variants of the density number.

Abstract

How many permutations are needed so that every infinite-coinfinite set of natural numbers with asymptotic density can be rearranged to no longer have the same density? We prove that the density number $\mathfrak{dd}$, which answers this question, is equal to the least size of a non-meager set of reals, $\mathsf{non} (\mathcal{M})$. The same argument shows that a slight modification of the rearrangement number $\mathfrak{rr}$ of~\cite{BBBHHL20} is equal to $\mathsf{non} (\mathcal{M})$, and similarly for a cardinal invariant related to large-scale topology introduced by Banakh~\cite{Ba23}, thus answering a question of the latter. We then consider variants of $\mathfrak{dd}$ given by restricting the possible densities of the original set and / or of the permuted set, providing lower and upper bounds for these cardinals and proving consistency of strict inequalities. We finally look at cardinals defined in terms of relative density and of asymptotic mean, and relate them to the rearrangement numbers of~\cite{BBBHHL20}.