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Fourier methods in additive combinatorics

作者:   时间:2022-06-14   点击数:

报告人:Xuancheng Shao   单位:University of Kentucky

题目:Fourier methods in additive combinatorics

时间:620日上午9:00开始

地点:Zoom会议

摘要:Fourier analysis is a fundamental tool in number theory, particularly in the detection of solutions to equations in arithmetically structured sets, as in Goldbach's conjecture and Szemeredi's theorem. This series of lectures explores the detection of arithmetic configurations within special sets of integers -- those for which we have only combinatorial information, such as the set of Chen primes and the set of primes in short intervals. Techniques originating in additive combinatorics, most notably the transference principle, have been developed to tackle these problems. The transference principle based on Fourier analysis was pioneered by Green, with further success in works of Green, Tao, and others. The intention of these lectures is to introduce these ideas in some model instances, with pointers towards more advanced implementations and open questions.

 

Lecture 1: Introduction, review of Fourier analysis 620日上午9:00开始)

Our motivation comes from solving equations in interesting sets. We start with recalling the method of Fourier analysis (or the Hardy-Littlewood circle method) to treat Vinogradov's three primes theorem. We then give an overview of related results in the literature concerning other equations and other interesting sets.

 

Lecture 2: The basic Fourier analytic transference principle 621日上午9:00开始)

In this lecture we prove Roth's theorem in Chen primes, a result due to Green and Tao, while highlighting the use of the transference principle via Fourier analysis.

 

Lecture 3: Variants of the transference principle and open questions 622日上午9:00开始)

In the final lecture we consider more advanced implementations of the basic transference principle. As an example, we describe the main ideas behind the proof of Vinogradov's theorem in Chen primes, a result due to Matomaki and myself. We will end with some open problems, concerning Piatetski-Shapiro primes, quadratic equations, primes in short intervals, etc.

 

邀请人:赵立璐

 

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