On recurrence for $\mathbb{Z}^d$-Weyl systems

Abstract

We study the topological recurrence phenomenon of actions of locally compact abelian groups on compact metric spaces. In the case of $\mathbb{Z}^d$-actions we develop new techniques to analyze Bohr recurrence sets. These techniques include finding and manipulating correlations between the coordinates of the set of recurrence. Using this, we show that Bohr recurrence sets are recurrence sets for $\mathbb{Z}^d$-Weyl systems. This family encompasses, for example, all $\mathbb{Z}^d$-affine nilsystems. To our knowledge, this is the first result towards a positive answer to Katznelson question in the case of $\mathbb{Z}^d$-actions.