一般格薄环中格点的统计
Statistics of lattice points in thin annuli for generic lattices
摘要：
本文研究了薄圆环内格点的计数函数的统计性质。根据Bleher和Lebowitz的猜想，如果宽度收缩到零，但面积收敛到无穷大，则分布收敛到高斯分布。如果宽度慢慢收缩到零，Hughes和Rudnick在标准格上证明了这个猜想，在我们之前的论文中也证明了一般的矩形格。我们证明了这个猜想的任意格满足一些通用丢番图性质，再次假设的宽度的annuli收缩缓慢为零。在这个问题上应用Hughes-Rudnick技术的障碍之一是存在所谓的密切对的格点。为了克服这一困难，我们绑定的发生率这种现象的扩展Eskin-Margulis-Mozes的工作的定量Openheim猜想。
英文摘要：
We study the statistical properties of the counting function of lattice points inside thin annuli. By a conjecture of Bleher and Lebowitz, if the width shrinks to zero, but the area converges to infinity, the distribution converges to the Gaussian distribution. If the width shrinks slowly to zero, the conjecture was proven by Hughes and Rudnick for the standard lattice, and in our previous paper for generic rectangular lattices. We prove this conjecture for arbitrary lattices satisfying some generic Diophantine properties, again assuming the width of the annuli shrinks slowly to zero. One of the obstacles of applying the technique of Hughes-Rudnick on this problem is the existence of so-called close pairs of lattice points. In order to overcome this difficulty, we bound the rate of occurence of this phenomenon by extending some of the work of Eskin-Margulis-Mozes on the quantitative Openheim conjecture.
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