Some problems in number theory

这个项目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，余将研究（1）有限集的整数满足一些二元加法性质的界限，和（2）秩0二次扭转的椭圆曲线的有理数。研究（1）的对象是整数的广义Sidon集和可加2-基。引入了一种新的思想，它捕获的和集和差集的浓度。沿着经典的离散傅里叶分析方法，这一新思想改进了目前广义Sidon集基数的最佳上界。这种新的方法也将被用来改善整数的2-基的下界。在研究（2）中，余文涛证明了，对于有理数上的任意椭圆曲线，其二次扭转的秩为0的比例为正;对于有理数上的一对椭圆曲线，存在无穷多个无平方因子的整数D，使得两条曲线被同一个D的二次扭转的秩同时为0。为了实现这些目标，将采用诸如第一下降法、筛选法和特征和估计等技术。此外，余承东将尝试用类似的方法证明有理数上每一条椭圆曲线的二次扭的平均秩的有界性。（广义）Sidon集的定界问题引起了人们的广泛关注.虽然广义Sidon集的上界在20世纪30年代Sidon的工作中对Fourier级数的研究有着自然的意义，但它也与调和分析和连续Ramsey理论中的其他一些问题密切相关。在本项目的第一部分中，Yu研究了广义Sidon集的上界，并利用其中的新思想研究了相关问题。在项目的另一部分，Yu致力于研究椭圆曲线的一些算法。Yu将部分解决一个与椭圆曲线秩有关的长期问题，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.