FU Berlin

Block Course – Towards the Polynomial Freiman-Ruzsa Conjecture

October 6 – 24 2014

Freie Universität Berlin

Topics and lecturers

Week 1: Giorgis Petridis (University of Rochester): Classical and new results in Additive Combinatorics
6 - 10 October 2014

Material: elementary inequalities in Additive Combinatorics (Ruzsa's triangle inequality), Plünnecke's inequality via Petridis proof, covering lemmas, Ruzsa's power trick, Additive energy and Balog-Szemerédi-Gowers Theorem, Freiman isomorphisms, sum-product estimates and Szemerédi-Trotter Theorem .

Handout 1, Handout 2, Handout 3, Handout 4, the full material and the Problems

Week 2: Julia Wolf (University of Bristol): Analytic techniques in Additive Combinatorics
13 - 16 October 2014

Material: introduction to the discrete Fourier transform and Bogolyubov’s Lemma, Meshulam’s Theorem, Croot-Sisask almost periodicity, applications including log-bound in Bogolyubov (finite fields), Roth’s theorem in the integers, Bohr sets and the notion of regularity with applications, introduction to higher-order uniformity


Week 3: Tomasz Schoen (Adam Mickiewicz University): The Polynomial Freiman-Ruzsa Conjecture
20 - 23 October 2014

Material: proof of Freiman Theorem (including some geometry of numbers, if needed, but it can be avoided even in the integers), Chang's spectral lemma, Schoen's attack on PFR, log-bound in Bogolyubov in the integers according to Sanders and Sanders attack (log-bound), consequences of effective bounds, new bound for size of sets without solution to a+b+c+d+e=5f (finite fields setting).

The precise schedule(lecture rooms, timetable, ...) of the Block Course


One of the first results in Additive Combinatorics is the so-called Freiman-Ruzsa Theorem, which states that a set with small sumset is in fact a subset of a generalized arithmetic progression. This theorem has emerged as an important cornerstone of the field, and many related questions are still unknown.

The course is intended for master students, PhD-students and postdoctoral researchers interested in the field of Additive Combinatorics and related areas. The course will start with a swift introduction to the classical results on the area (Week 1). Later, on Week 2, the course will move to Fourier Analytic techniques and applications in Additive Combinatorics. Finally, in Week 3, Freiman-Ruzsa Theorem and the latest achievements towards the Polynomial Freiman-Ruzsa Conjecture will be discussed.

Participants can join any part of the course, depending on their individual background and interest.


The course is addressed to graduate students and postdocs of Mathematics or Computer Science, who are interested in additive combinatorics, applications of Fourier analysis in discrete mathematics, and related fields.


The Block Course is organized within the Research Training Group "Methods for Discrete Structuresand organized by Jun. Prof. Juanjo Rué and Prof. Tibor Szabó.


Applications for participation in the course (the whole or part of it -- please indicate the weeks you plan to participate in), with a short curriculum vitae and scientific background, should be sent by
June 30, 2014, preferably by email, to Jun. Prof. Juanjo Rué at jrue (at)

Through the generous support of the Department of Mathematics and Computer Science of Freie Universität Berlin and the Research Training Group "Methods for Discrete Structures, there is a limited amount of financial support available for PhD-students or for advanced Master/Diploma students in a field related to the topics of the course. Applications for financial support, with curriculum vitae, copies of certificates, thesis, areas of interest, and a letter of recommendation (sent directly by the letter writer) should be sent by May 31, 2014, preferably by e-mail, to Jun. Prof. Juanjo Rué at jrue (at)

Practical Hints

This is the place where you find hints on how to find reasonably priced accomodation, how to get around Berlin, and the like. If there is anything missing, let us know. (Some of the links lead to German pages, sorry).