Effective equidistribution of primitive rational points on expanding horospheres

2022/12/14 by Daniel El-Baz, Min Lee, El-Baz, Daniel +2
Mathematics · #Analytic Number Theory Research #Advanced Algebra and Geometry #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2212.07408

Abstract

We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira on the asymptotic distribution of primitive rational points on expanding closed horospheres in the space of lattices. Key ingredients of our proof include recent bounds on matrix Kloosterman sums due to Erdélyi and Tóth, results by Clozel, Oh and Ullmo on the effective equidistribution of Hecke points, and Rogers' integration formula in the geometry of numbers. As an application of the main theorem, we also obtain a result on the limit distribution of the number of small solutions of a random system of linear congruences to a large modulus. Furthermore, as a by-product of our proofs, we obtain a sharp bound on the number of nonsquare matrices over a finite field \mathbbF_p with small entries and of a given size and rank.

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