Cardinal characteristics associated with small subsets of reals

Abstract

Inspired by Bartoszy\'nski's work on small sets, we introduce a new ideal defined by interval partitions on natural numbers and summable sequences of positive reals. Similarly, we present another ideal that relies on Bartoszy\'nski's and Shelah's representation of $F_\sigma$ measure zero sets. We show they are $\sigma$-ideals characterizing all small sets and $F_\sigma$ measure zero sets. We also study the cardinal characteristics associated with the introduced ideals. We use them to describe the invariants of measure, discuss their connection to Cicho\'n's diagram, and present related consistency results.