Submission date: 25 October 2022

Abstract:

We introduce a relaxation of stability, called robust stability, which is insensitive to perturbations by subsets of Loeb measure $0$ in a pseudofinite group. We show that robust stability satisfies a stationarity principle for measure independent elements. We apply this principle to deduce the existence of squares and $L$-shapes in dense subsets of Cartesian squares of pseudofinite groups, possibly non-abelian. Our results imply qualitative asymptotic versions for Cartesian squares of finite groups.

Mathematics Subject Classification: 03C13, 03C45, 11B30

Keywords and phrases:

Full text arXiv 2210.14039: pdf, ps.