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Stability, corners, and other two-dimensional shapes
- Part of
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 26 December 2025, pp. 1-29
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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 two-dimensional subsets of finite abelian groups of odd order.
