Mean value theorems for the S-arithmetic primitive Siegel transforms

2023/10/05 by Samantha Fairchild, Jiyoung Han, Fairchild, Samantha +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2310.03459

openalex publication_date 2023/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop the theory and properties of primitive unimodular S-arithmetic lattices in ℚ_S^d by giving integral formulas in the spirit of Siegel's primitive mean value formula and Rogers' and Schmidt's second moment formulas. When d=2, unlike in the real case, functions arising from the S-primitive Siegel transform are unbounded, requiring a careful analysis to establish their integrability. We then use mean value and second moment formulas in three applications. First, we obtain quantitative estimates for counting primitive S-arithmetic lattice points. We next establish a quantitative Khintchine--Groshev theorem, which, in the real case, involves counting primitive integer points in ℤ^d subject to congruence conditions. Finally, we derive an S-arithmetic logarithm law for unipotent flows in the spirit of Athreya--Margulis. These applications follow the spirit of the real case, but require new technical aspects of the proofs, particularly when d=2.

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