关于二维格点间平行轴空矩形的面积
On the area of empty axis-parallel rectangles amidst 2-dimensional lattice points
摘要：
单位正方形中点集的离散度定义为最大空轴平行盒的面积。在本文中，我们感兴趣的是在平面上的分散的格，即，在格点之间的空轴平行盒的面积的上确界。我们引入了一个框架，研究这一点的基础上的连分数扩展的发电机的格。这个框架证明是如此成功，以至于我们无法提出一个我们无法回答的问题。给出了格具有有限色散的充要条件。得到了二次域的整数环的子群所对应的格的色散的一个精确公式。基于生成元的最大连分数系数，我们对格的色散有严格的限制，精确到一半以内。我们知道第$n$最好的格是什么。我们提供了一个等价的Zaremba猜想的制定。使用我们的框架，我们能够给其他两篇论文的结果在短短几行证明。
英文摘要：
The dispersion of a point set in the unit square is defined to be the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices in the plane, that is, the supremum of the area of the empty axis-parallel boxes amidst the lattice points. We introduce a framework with which to study this based on the continued fractions expansions of the generators of the lattice. This framework proves so successful that we were unable to ask a question that we could not answer. We give necessary and sufficient conditions under which a lattice has finite dispersion. We obtain an exact formula for the dispersion of the lattices associated to subgroups of the ring of integer of a quadratic field. We have tight bounds for the dispersion of a lattice based the largest continued fraction coefficient of the generators, accurate to within one half. We know what the $n$-th best lattice is. We provide an equivalent formulation of Zaremba's conjecture. Using our framework we are able to give alternative proofs of the results from two other papers in only a few lines.
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