Some problems in number theory

本课题的目标是研究组合数论和解析数论中的一些经典问题。更确切地说，Yu将研究(1)满足某些二元可加性的有限整数集的界，以及(2)有理数上椭圆曲线的0阶二次扭转。研究(1)的对象是广义西顿集和整数的可加2基。引入了一种新的思想，即捕捉集和差集的集中。与经典的离散傅里叶分析方法一起，这一新的思想改进了目前广义西顿集的基数的最佳上界。这种新方法也将用于改进整数的2基下界。在研究(2)中，Yu将证明，对于任何在有理数上具有2-等根的椭圆曲线，其二次扭转有正比例为0；并且，对于一对有理数上的椭圆曲线，存在无限多个无平方整数D，使得两条曲线同时被同一个D所二次扭转的秩为0。为了实现这些，将应用诸如第一次下降、筛选方法和字符和估计等技术。此外，Yu将尝试用类似的方法证明每条椭圆曲线的二次扭转在有理数上的平均秩的有界性。（广义）西顿集的边界问题引起了人们的广泛关注。广义西顿集的上界在20世纪30年代西顿的傅立叶级数研究中有着自然的蕴涵，它也与谐波分析和连续拉姆齐理论中的其他一些问题密切相关。在本课题的第一部分，Yu研究了广义Sidon集的上界，并利用所涉及的新思想研究了一些相关问题。在项目的另一部分，Yu致力于研究椭圆曲线的一些算法。Yu将部分解决与椭圆曲线秩相关的长期问题，并将研究一些相关问题。

The goal of this project is to study some classic problems in combinatorial and analytic number theory. More precisely, Yu will study (1) bounds for finite sets of integers satisfying some binary additive properties, and (2) rank 0 quadratic twists of elliptic curves over the rationals. The objects in study (1) are the generalized Sidon sets and additive 2-basis of integers. A new idea is introduced which captures the concentration of the sumset and the difference set. Along with the classic methods of discrete Fourier analysis, this new idea results in an improvement for currently the best upper bounds for the cardinalities of generalized Sidon sets. This new approach will also be employed to improve the lower bound for the 2-basis of integers. In study (2), Yu will show that, for any elliptic curve which posses a 2-isogeny over the rationals, a positive proportion of its quadratic twists have rank 0; and, for a pair of elliptic curves over the rationals, there are infinitely many squarefree integers D such that the quadratic twists of the two curves by the same D simultaneously have rank 0. To achieve these, techniques such as the first descent, sieve methods and estimation of character sums will be applied. Besides, Yu will try to use the similar methods to prove boundedness of the average rank of the quadratic twists of every elliptic curve over the rationals. The problem of bounding a (generalized) Sidon set has attracted a lot of attentions. While an upper bound for a generalized Sidon set has natural implication in the study of Fourier series from Sidon's work back in 1930's, it is also closely related to some other problems in harmonic analysis and continuous Ramsey theory. In the first part of this project, Yu studies the upper bounds for generalized Sidon sets, and use the new idea involved to study some related problems. In the other part of the project, Yu is devoted to studying some arithmetic of elliptic curves. Yu will partially solve a longstanding problem related to the rank of elliptic curves, and will also study some related problems.