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arXiv · 2505.19413

Limit distributions for $\text{SO}(n,1)$ action on $k$-lattices in $\mathbb{R}^{n+1}$

Michael Bersudsky·Nimish A. Shah
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Abstract

We study the asymptotic distribution of norm ball averages along orbits of a lattice $\Gamma \subset \text{SO}(n,1)$ acting on the moduli space of pairs of orthogonal discrete subgroups of $\mathbb{R}^{n+1}$ up to homothety. Our main result shows that, except for special $2$-lattices in $\mathbb{R}^3$ lying in hyperplanes tangent to the light cone, these measures converge to an explicit semi-invariant probability measure supported on the space of homothety classes of pairs of orthogonal lattices tangent to the light cone. Our main motivation is a conjecture of Sargent and Shapira, which is resolved as a special case of our general result.

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Michael Bersudsky, Nimish A. Shah. 2025-05-26. Limit distributions for $\text{SO}(n,1)$ action on $k$-lattices in $\mathbb{R}^{n+1}$. https://arxiv.org/abs/2505.19413

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