Small-large subgroups of the reals

TL;DR

This paper investigates subgroups of the real numbers that are simultaneously small in one measure or category sense and large in another, revealing their existence and consistency results within ZFC.

Contribution

It demonstrates the existence of a non-meager Lebesgue null subgroup of R in ZFC and shows the consistency of the non-existence of a non-null meager subgroup.

Findings

Existence of a non-meager Lebesgue null subgroup in ZFC

Consistency of no non-null meager subgroup of R

Results depend on set-theoretic assumptions

Abstract

We are interested in subgroups of the reals that are small in one and large in another sense. We prove that, in ZFC, there exists a non-meager Lebesgue null subgrooup of R, while it isconsistent there there is no non-null meager subgroup of R.