A Szemeredi type theorem for sets of positive density in approximate lattices

Artikel i vetenskaplig tidskrift, 2024

An extension of Szemeredi's theorem is proved for sets of positive density in approximate lattices in general locally compact and second countable abelian groups. As a consequence, we establish a recent conjecture of Klick, Strungaru and Tcaciuc. Via a novel version of Furstenberg's correspondence principle, which should be of independent interest, we show that our Szemer & eacute;di theorems can be deduced from a general transverse multiple recurrence theorem, which we establish using a recent work of Austin [Non-conventional ergodic averages for several commuting actions of an amenable group. J. Anal. Math.130 (2016), 243-274].