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arXiv · 2401.03724

Simplices in large sets and directional expansion in ergodic actions

Michael Björklund·Alexander Fish
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Abstract

In this paper we study ergodic $\mathbb{Z}^r$-actions and investigate expansion properties along cyclic subgroups. We show that under some spectral conditions there are always directions which expand significantly a given measurable set with positive measure. Among other things, we use this result to prove that the set of volumes of all $r$-simplices with vertices in a set with positive upper density must contain an infinite arithmetic progression, thus showing a discrete density analogue of a classical result by Graham.

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Michael Björklund, Alexander Fish. 2024-01-08. Simplices in large sets and directional expansion in ergodic actions. https://doi.org/10.1017/fms.2024.125

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