A Szemer\'edi type theorem for sets of positive density in approximate lattices

An extension of Szemer\'edi's Theorem is proved for sets of positive density in cut-and-project sets in general locally compact and second countable abelian groups. As a consequence, we establish a recent conjecture of Klick, Strungaru and Tcaciuc. Furthermore, we show a Szemeredi type theorem for sets of positive density in the set of fractional parts of Qₚ. Via a novel version of Furstenberg's Correspondence principle, which should be of independent interest, we show that our Szemer\'edi Theorems can be deduced from a general transverse multiple recurrence theorem, which we establish using recent works of Austin.