Low-complexity Haar null sets without $G_\delta $ hulls in $\mathbb {Z}^\omega $
Volume 246 / 2019
Fundamenta Mathematicae 246 (2019), 275-287
MSC: Primary 03E15; Secondary 28C10, 22F99.
DOI: 10.4064/fm628-9-2018
Published online: 9 May 2019
Abstract
We show that for every $2\le \xi \lt \omega _1$ there exists a Haar null set in $\mathbb {Z}^\omega $ that is the difference of two $\mathbf {\Pi }^0_\xi $ sets but is 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 is 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 $.
