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 $$\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> actions, we develop new techniques to analyze Bohr recurrence sets, and show that Bohr recurrence implies recurrence for $$\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> -Weyl systems. This family encompasses, for example, all $$\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> -affine nilsystems.