Limit distributions for SO(n,1) action on k-lattices in ℝⁿ⁺¹

2025/05/26 by Michael Bersudsky, Bersudsky, Michael, Nimish A. Shah +1
Mathematics · #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2505.19413

openalex publication_date 2025/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the asymptotic distribution of norm ball averages along orbits of a lattice Γ⊂ SO(n,1) acting on the moduli space of pairs of orthogonal discrete subgroups of ℝⁿ⁺¹ up to homothety. Our main result shows that, except for special 2-lattices in ℝ³ 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.

Citations

• Dynamics on the space of 2-lattices in 3-space
• Counting lattice points
• Distribution of lattice orbits on homogeneous varieties
• Limit distributions of expanding translates of certain orbits on homogeneous spaces

Related

• Limiting distribution of dense orbits in a moduli space of rank m discrete subgroups in (m+1)-space
• Equidistribution of lattice orbits in the space of homothety classes of rank 2 sublattices in \mathbb R³
• Distribution of orbits of geometrically finite groups acting on null vectors
• Distribution of lattice orbits on homogeneous varieties
• Best possible rates of distribution of dense lattice orbits in\n homogeneous spaces