Stability, corners, and other 2-dimensional shapes

Abstract

We introduce a relaxation of stability, called almost sure stability, which is insensitive to perturbations by subsets of Loeb measure $0$ in a non-standard finite group. We show that almost sure stability satisfies a stationarity principle in the sense of geometric stability theory for measure independent elements. We apply this principle to deduce the existence of squares in dense almost surely stable subsets of Cartesian products of non-standard finite groups, possibly non-abelian. Our results imply qualitative asymptotic versions for Cartesian products of finite groups. In the final section, we establish the existence of $3\times 2$-grids (and thus of $L$-shapes) in dense almost surely stable $2$-dimensional subsets of finite abelian groups of odd order.