Some problems in number theory

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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.