Star operation, microscopic sets and porous sets
Daria Perkowska, Szymon Żeberski
TL;DR
This work investigates the star operation on families of subsets in Cantor space and its connections to porous and microscopic sets within the framework of small-set theory. It extends the Galvin–Mycielski–Solovay theorem to $2^{\omega}$, analyzing strong measure zero and strongly meager sets, and examines how star-closure relates to countable-ideal constructions and Borel-type conjectures. The paper provides explicit constructions showing when $\mathcal{F}=\mathcal{F}^{*}$ and when $\mathcal{F}=\mathcal{F}^{**}$ fail or hold, highlighting delicate interactions between $\mathcal{M}, \mathcal{N}, Count$, and translation-invariant $\sigma$-ideals. It further develops Cantor-space analogues of porosity and microscopic sets, establishing fundamental inclusions, presenting counterexamples that separate porous from microscopic hierarchies, and proving a key lemma that one direction of the SMZ–meager interaction extends to porous and microscopic contexts. Overall, the results illuminate how algebraic sums, ideals, and notions of smallness interact in Cantor space, with implications for the structure and consistency of BC-like conjectures.
Abstract
This paper explores the interplay between star operations, microscopic sets, and porous sets. The study focuses on the Galvin-Mycielski-Solovay theorem, which characterizes strongly measure zero sets and their interactions with meager sets. Results include the investigation of the star operation $\mathcal{F}^*$ and its properties. The paper also examines the relationship between porous sets and microscopic sets. Additionally, the work presents constructions of families $\mathcal{F}$ in $\mathcal{P}(\mathbb{Z}), \mathcal{P}(\mathbb{Z}^ω),$ and $\mathcal{P}(2^ω)$ that satisfy $\mathcal{F} = \mathcal{F}^*$. Theorems and lemmas are provided to establish conditions under which $\mathcal{F}^{**} = \mathcal{F}$ and to analyze the implications of the Borel Conjecture and its dual. The paper concludes with a discussion of microscopic sets and their properties, including their interactions with porous sets and the non-equivalence of certain classes of sets.