论文标题
通过模块化在圆环上分发点
Distributing Points on the Torus via Modular Inverses
论文作者
论文摘要
我们研究了有关点的分布\ [\ [\ left \ {\ left（\ frac {d} {q}，\ frac {\ overline {d}} {q} {q} {q} \ right）\ in \ mathbb {t}^2：d d \ in （\ Mathbb {Z}/Q \ Mathbb {Z}）^{\ times} \ right \} \] a $ q $倾向于无限。由于Kloosterman总和的非平凡界限，众所周知，这些点在圆环上。我们证明了此结果的改进，包括差异的界限，与这些点相关的覆盖指数的范围，稀疏等分分配和混合。
We study various statistics regarding the distribution of the points \[\left\{\left(\frac{d}{q},\frac{\overline{d}}{q}\right) \in \mathbb{T}^2 : d \in (\mathbb{Z}/q\mathbb{Z})^{\times}\right\}\] as $q$ tends to infinity. Due to nontrivial bounds for Kloosterman sums, it is known that these points equidistribute on the torus. We prove refinements of this result, including bounds for the discrepancy, small scale equidistribution, bounds for the covering exponent associated to these points, sparse equidistribution, and mixing.
