Mathematical Sciences: Probabilistic Diophantine Approximation

PI将研究数论中几个不同但相互联系的问题。第一部分将重点讨论非齐次Pell不等式的积分解的渐近性质。该建议的另一部分涉及对刚刚描述的部分的全局版本的研究，即计算倾斜矩形中晶格点的数量。这是数论领域的研究。数论从整数和诸如整数能被整数整除之类的问题开始。它是数学中最古老的领域之一，最初纯粹出于审美原因而追求它。然而，在过去的半个世纪里，它已经成为开发计算机科学新算法和电子学新纠错码的重要工具。

The PI will investigate several different, yet mutually connected problems in number theory. One part will focus on the asymptotic behavior of integral solutions to inhomogeneous Pell inequalities. Another part of this proposal deals with the study of the global version to the part just described, namely, counting the number of lattice points in tilted rectangles. This is research in the field of number theory. Number theory starts with the whole numbers and questions such as the divisibility of one whole number by another. It is among the oldest fields of mathematics and it was originally pursued for purely aesthetic reasons. However, within the last half century, it has become an essential tool in developing new algorithms for computer science and new error correcting codes for electronics.