Lecture 2: Density Increment, Fourier Methods in Combinatorial Number Theory

As part of the LMS Scheme 3 Covid response, we are hosting a series of online lectures on 'Fourier methods in combinatorial number theory'. The Scheme 3 Covid response aims to provide extra training and support to early career researchers during the pandemic.

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Description: Fourier analysis is a fundamental tool in number theory, particularly in the detection of solutions to equations in arithmetically structured sets, as in Waring's problem or Goldbach's conjecture. This course explores the detection of arithmetic configurations within unstructured sets of integers - those for which we have only combinatorial information, such as their density. Techniques originating in additive combinatorics have been developed to tease out the latent structure present in such sets, separating it from random 'noise'. Resolving these combinatorial problems can then feed back into more classical problems of number theory, such as detecting arithmetic structures in the primes.  The Fourier-analytic approach to Szemerédi-type problems was pioneered by Roth and Gowers, with spectacular further success in work of Green, Tao and Ziegler on patterns in the primes. The intention of this course is to introduce these ideas in their simplest instances, with pointers towards more advanced implementations and quantitative refinements. 

Lecturer: Sean Prendiville

Intended audience: Graduate students in number theory, combinatorics and analysis.

Pre-requisites: Nothing beyond undergraduate analysis and a fortitude for persevering with epsilon-delta management.

Key words: additive combinatorics; higher order Fourier analysis; Szemerédi's theorem; combinatorial number theory; the Hardy-Littlewood circle method. 

Syllabus: Heuristics in the circle method; counting in Fourier uniform sets; density increment; the arithmetic regularity lemma; energy increment; dual functions; the transference principle.

Dates: Wednesday 4pm (GMT, London); November 4, 11, 18, 25; December 2, 9.

All lectures will be broadcast live via Zoom and uploaded to YouTube after the event.

Lecture notes (updated the week before each lecture): https://bit.ly/3l1cVA2