Constructing universally small subsets of a given packing index in Polish groups

TL;DR

This paper constructs universally small subsets in uncountable Abelian Polish groups with specified intersection and packing properties, under the Continuum Hypothesis, advancing understanding of small sets in descriptive set theory.

Contribution

It introduces a method to construct universally small subsets with prescribed packing indices in Abelian Polish groups under CH, a novel approach in the field.

Findings

Existence of universally small sets with large intersection properties.

Construction of universally small sets with arbitrary sharp packing index.

Results hold under the Continuum Hypothesis.

Abstract

A subset of a Polish space $X$ is called universally small if it belongs to each ccc $\sigma$-ideal with Borel base on $X$. Under CH in each uncountable Abelian Polish group $G$ we construct a universally small subset $A_0\subset G$ such that $|A_0\cap gA_0|=\mathfrak c$ for each $g\in G$. For each cardinal number $\kappa\in[5,\mathfrak c^+]$ the set $A_0$ contains a universally small subset $A$ of $G$ with sharp packing index $\pack^\sharp(A_\kappa)=\sup\{|\mathcal D|^+:\mathcal D\subset \{gA\}_{g\in G}$ is disjoint$\}$ equal to $\kappa$.