Cardinal characteristics associated with small subsets of reals

TL;DR

This paper introduces new ideals related to small sets of reals, explores their cardinal characteristics, and connects these invariants to measure theory and Cichoń's diagram, providing new insights and consistency results.

Contribution

It defines novel ideals based on interval partitions and summable sequences, characterizes small sets and $F_\sigma$ measure zero sets, and analyzes their cardinal invariants and relationships.

Findings

New ideals characterize small sets and measure zero sets.

Cardinal invariants related to these ideals are studied.

Connections to Cichoń's diagram and consistency results are established.

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.