Fourier Analysis and Arithmetic Progressions

Fourier analysis has become one of the most important tools in additive combinatorics and it is behind many important results in the area. The purpose of this mini-course is to present the use of discrete Fourier analysis with combinatorial applications in mind. We will introduce the subject from scratch with the aim of covering the recent breakthrough of Kelley and Meka on 3-term arithmetic progressions in dense subsets of integers.

A tentative outline of the course is as follows:

• Fourier analysis on finite abelian groups
• Roth’s theorem and Behrend’s construction
• The finite field model and Bohr sets
• Dependent random choice and almost periodicity
• Fourier positivity and the Kelley–Meka bound

Time permitting, we will explore further applications of these ideas. Lecture notes and exercises will be provided.

References:
[1] F.A. Behrend, On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci., 32 (1946), 331–332.
[2] T.F. Bloom and O. Sisask, The Kelley–Meka bounds for sets free of three-term arithmetic progressions, arXiv:2302.07211.
[3] T.F. Bloom and O. Sisask, An improvement to the Kelley–Meka bounds on three-term arithmetic progressions, arXiv:2309.02353.
[4] Z. Kelley and R. Meka, Strong bounds for 3-progressions, arXiv:2302.05537.
[5] T. Schoen and O. Sisask, Roth’s theorem for four variables and additive structures in sums of sparse sets, Forum Math. Sigma, 4 (2016), e5.