Geodesic
Simplices in large sets and directional expansion in ergodic actions
Forum of Mathematics, Sigma, Tome 12 (2024)

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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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     author = {Michael Bj\"orklund and Alexander Fish},
     title = {Simplices in large sets and directional expansion in ergodic actions},
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Michael Björklund; Alexander Fish. Simplices in large sets and directional expansion in ergodic actions. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.125

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