论文标题
在扩展霍斯磷的原始理性点的有效等分
Effective equidistribution of primitive rational points on expanding horospheres
论文作者
论文摘要
由于Einsiedler，Mozes，Shah和Shapira在晶格范围内扩展闭合的hOspheres时，我们证明了结果的有效版本。我们证明的关键要素包括由于Erdélyi和Tóth引起的矩阵Kloosterman总和的最新界限，Clozel，OH和Ullmo的结果是Hecke点的有效等分分配，以及Rogers在数量几何学中的集成公式。作为主定理的应用，我们还获得了一个结果，即对大型模量的线性一致性随机系统的小解决方案的极限分布。此外，作为我们证明的副产品，我们在有限字段$ \ mathbb {f} _p $带有小条目以及给定尺寸和等级的有限字段$ \ mathbb {f} _p $上获得了尖锐的界限。
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 $\mathbb{F}_p$ with small entries and of a given size and rank.
