Borel's conjecture and meager-additive sets

TL;DR

This paper demonstrates the relative consistency of a set-theoretic model where all strong measure zero sets are meager-additive, yet uncountable strong measure zero sets exist, addressing a longstanding open question.

Contribution

It shows the consistency of Borel's conjecture failing while all strong measure zero sets are meager-additive, resolving a question posed by Bartoszyński and Judah.

Findings

Every strong measure zero set can be meager-additive in some models.

Uncountable strong measure zero sets can exist even when all such sets are meager-additive.

The result is consistent with ZFC, indicating independence from standard set theory.

Abstract

We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a long-standing question due to Bartoszy\'nski and Judah.