Non-meagre subgroups of reals disjoint with meagre sets

TL;DR

The paper proves that in certain groups, there exist large subgroups without the Baire property that are disjoint from translations of meagre sets, highlighting differences between category and measure.

Contribution

It establishes the existence of non-meagre subgroups disjoint from meagre set translations in specific groups, extending previous techniques.

Findings

Existence of subgroups without the Baire property disjoint from meagre sets

Extension of Rosłanowski and Shelah's proof techniques

Contrast with measure-theoretic analogues in ZFC

Abstract

Let $(X, +)$ denote $(\mathbb{R}, +)$ or $(2^{\omega}, +_2)$. We prove that for any meagre set $F \subseteq X$ there exists a subgroup $G \le X$ without the Baire property, disjoint with some translation of F. We point out several consequences of this fact and indicate why analogous result for the measure cannot be established in ZFC. We extend proof techniques from the work of Ros{\l}anowski and Shelah [1].