Additivity of the ideal of microscopic sets

TL;DR

The paper investigates the properties of microscopic sets, showing their additivity is  and providing new insights into their covering properties and related ideals within set theory.

Contribution

It proves that the additivity of the microscopic sets' ideal is , solving a problem posed by G. Horbaczewska, and explores related generalizations.

Findings

Existence of a microscopic set not coverable by a sequence with lower asymptotic density zero.

Additivity of the microscopic sets' ideal is  in ZFC.

Discussion of additivity for generalized microscopic ideals.

Abstract

A set $M\subset\mathbb{R}$ is microscopic if for each $\varepsilon>0$ there is a sequence of intervals $(J_n)_{n\in\omega}$ covering $M$ and such that $|J_n|\leq \varepsilon^{n+1}$ for each $n\in\omega$. We show that there is a microscopic set which cannot be covered by a sequence $(J_n)_{n\in\omega}$ with $\{n\in\omega:J_n\neq\emptyset\}$ of lower asymptotic density zero. We prove (in ZFC) that additivity of the ideal of microscopic sets is $\omega_1$. This solves a problem of G. Horbaczewska. Finally, we discuss additivity of some generalizations of this ideal.