Mathematical Sciences: "Hardy-Littlewood System"

本研究是根据Hard和Littlewood的圆法的预测，了解仿射变量上有理点的渐近分布。第一部分讨论一般变分，提出了Hardy-Littlewood变分的概念，并定义了测量这种变分与满足Hardy-Littlewood型的精确渐近的偏差的不变量。发现了各种Hardy-Littlewood存在的代数-几何和算术性质的障碍。另一部分研究了半单群的齐次空间的例子，将算术群轨道上点的密度的结果与质量公式的版本相结合，得到了Hardy-Littlewood型的渐近结果。这项研究是在代数几何领域进行的，代数几何是现代数学中最古老的部分之一，但它已经发展到解决了几个世纪以来一直存在的问题的地步。最初，它用最简单的方程，即多项式来处理平面上定义的图形。今天，该领域不仅使用代数的方法，而且还使用分析和拓扑的方法，相反，它在这些领域中被广泛使用。＊＊＊

Rudnick 9400163 This research is concerned with understanding the asymtotic distribution of rational points on affine varieties in accordance with predictions based on the circle of method of Hard and Littlewood. One part deals with general varieties, proposes a notion of Hardy-Littlewood variety and defines an invariant for measuring the deviation of such a variety from satisfying a precise asymptotic of Hardy-Littlewood type. Obstructions of algebro-geometric and arithmetic nature to a variety being Hardy-Littlewood are found. The other part is concerned with studying examples which are homogeneous spaces for semisimple groups, by combining results about the density of points in orbits of arithmetic groups with versions of mass formulae to obtain asymptotic results of Hardy-Littlewood type. The research is in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures defined in the plane by the simplest of equations, namely polymonials. Today, the field uses methods not only from algebra, but also from analysis and topology, and conversely it is extensively used in those fields. ***