Low complexity Haar null sets without G_δ hulls in Z^ω

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 ${\mathbf{\Pi }}_{\xi }^0$ sets but not contained in any ${\mathbf{\Pi }}_{\xi }^0$ 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. Vidnyánszky. To prove this, we also prove a theorem which characterizes the Haar null subsets of ${\mathbb{Z}}^{\omega }$.