A Polish group containing a Haar null $F_\sigma$-subgroup that cannot be enlarged to a Haar null $G_\delta$-set

TL;DR

This paper constructs a specific Polish meta-abelian group with a Haar null $F_\sigma$-subgroup that cannot be extended to a Haar null $G_\delta$-set, answering a question in the field.

Contribution

It provides the first example of such a group and subgroup, demonstrating a new phenomenon in the structure of Haar null sets in Polish groups.

Findings

Constructed a Polish meta-abelian group with the desired properties

Showed the subgroup cannot be enlarged to a Haar null $G_\delta$-set

Answered an open question by Elekes and Vidnyánszky

Abstract

Answering a question of Elekes and Vidny\'anszky, we construct a Polish meta-abelian group $H$ and a subgroup $F\subset H$, which is a Haar null $F_\sigma$-set in $H$ that cannot be enlarged to a Haar null $G_\delta$-set.