Cardinal invariants associated with the combinatorics of the uniformity number of the ideal of meager-additive sets

Abstract

In [CMRM24], it was proved that it is relatively consistent that \emph{bounding number} $\mathfrak{b}$ is smaller than the uniformity of $\mathcal{MA}$, where $\mathcal{MA}$ denotes the ideal of the meager-additive sets of $2^{\omega}$. To establish this result, a specific cardinal invariant, which we refer to as $\mathfrak{b}_b^\mathsf{eq}$, was introduced in close relation to Bartoszy\'nski's and Judah's characterization of the uniformity of $\mathcal{MA}$. This survey aims to explore this cardinal invariant along with its dual, which we call as $\mathfrak{d}_b^\mathsf{eq}$. In particular, we will illustrate its connections with the cardinals represented in Cicho\'n's diagram. Furthermore, we will present several open problems pertaining to these cardinals.