Stability, corners, and other 2-dimensional shapes

TL;DR

The paper introduces almost surely stability as a robust relaxation of classical stability for relations on Cartesian products of non-standard finite groups endowed with the Loeb measure $\mu_G$. A stationarity principle for almost stable relations is established via model-theoretic tools, notably the stabilizer $G_{M}^{00}$, enabling qualitative density results for two-dimensional configurations. Using this framework, the authors prove the existence of squares in dense almost surely stable subsets (and, in abelian groups, the existence of dense squares), and extend the analysis to $3\times 2$ grids and $L$-shapes, including in abelian odd-order groups. These results yield finitary, density-based corollaries for large finite groups and provide a bridge between model theory and additive combinatorics, with applications to non-abelian settings and to asymptotic questions in Cartesian products of finite groups.

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.

Theorems & Definitions (68)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.5
• Remark 2.6
• Lemma 2.7
• proof
• Remark 2.8
• Remark 2.9
• Definition 3.1