Low complexity Haar null sets without G_{\delta} hulls in Z^\omega

TL;DR

The paper constructs specific Haar null sets in  that are differences of certain definable sets but not contained in any smaller class of such sets, addressing a question in descriptive set theory.

Contribution

It demonstrates the existence of Haar null sets with particular definability properties that challenge previous assumptions, and characterizes Haar null subsets in .

Findings

Existence of Haar null sets as differences of _ sets not contained in any _ Haar null set.

Construction of Haar null sets that are differences of two G_ sets but not contained in any G_ Haar null set.

Characterization theorem for Haar null subsets of .

Abstract

We show that for every $2\le \xi<\omega_1$ there exists a Haar null set in $\mathbb{Z}^\omega$ that is the difference of two $\mathbf{\Pi}^0_\xi$ sets but not contained in any $\mathbf{\Pi}^0_\xi$ Haar null set. In particular, there exists a Haar null set in $\mathbb{Z}^\omega$ that is the difference of two $G_\delta$ sets but not contained in any $G_\delta$ Haar null set. This partially answers a question of M. Elekes and Z. Vidny\'anszky. To prove this, we also prove a theorem which characterizes the Haar null subsets of $\mathbb{Z}^\omega$.