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arXiv · 2402.17158

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

Michael Björklund·Alexander Fish
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Abstract

An extension of Szemer\'edi'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\'edi Theorems can be deduced from a general \emph{transverse} multiple recurrence theorem, which we establish using recent works of Austin.

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Michael Björklund, Alexander Fish. 2024-02-27. A Szemer\'edi type theorem for sets of positive density in approximate lattices. https://doi.org/10.1017/etds.2024.128

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