The course has little prerequisites beyond basic analysis and linear algebra. Some knowledge of integration theory, probability, and of the theory of finite fields, can be useful.
A DMATH forum is available for discussion and to ask/answer questions about the course, see here.
I will prepare lecture notes during the course of the semester.
The various drafts will appear here, with a date indicating which
version it is.
Version
of 20.12.2023.
The lecture will be accompanied by roughly biweekly exercise classes, usually during the Thursday class. We will announce the precise dates in the lecture as well as here. You should submit your exercise sheets either in person in the Tuesday lecture before the next exercise class or in PDF form to the coordinator: cynthia.bortolotto [at] math.ethz.ch.
Dates of exercise classes |
---|
September 28 |
October 12 |
October 26 |
November 9 |
November 23 |
December 7 |
Exercise sheet | Due by | Solutions |
---|---|---|
Exercise sheet 1 | October 3 | Exercise sheet 1 |
Exercise sheet 2 | October 10 | Exercise sheet 2 |
Exercise sheet 3 | October 26 | Exercise sheet 3 |
Exercise sheet 4 | November 9 | Exercise sheet 4 |
Exercise sheet 5 | November 23 | Exercise sheet 5 |
Exercise sheet 6 | December 7 | Exercise sheet 6> |
Day | Content |
---|---|
19.9.2023 | Introduction: what is additive combinatorics? Some basic statements, and some applications. Outline of the contents of the course. Review of some basic facts. |
21.9.2023 | Survey of Fourier analysis on finite abelian groups. |
26.9.2022 | Definition and basic properties and examples of product sets and Freiman morphisms. Definition of Sidon sets, application to the fourth moment; the Erdös-Turán example. |
3.10.2023 | The Eberhard--Manners classification of "classical" dense Sidon sets. Statements of basic results concerning Sidon sets (finite and infinte) in positive integers. |
5.10.2023 | Proof of upper bound for the "density" of infinite Sidon sets in positive integers. Definition of approximate subgroups and first quick discussion. |
10.10.2023 | Approximate subgroups: the Ruzsa distance, covering lemma, and control of higher product sets from the triple product set. Link between approximate subgroups and neutral symmetric sets with small tripling. Definition of multiplicative energy and statement of the link between sets with large normalized energy and approximate subgroups. |
17.10.2023 | A sample application of the idea of approximate subgroups: the Bourgain-Gamburd on expander graphs associated to SL_{2}(F_{p}); definition of expanders, and sketch of the key steps. |
19.10.2023 | Proof of a version of the Balog-Szemerédi-Gowers Theorem (following Schoen). Short discussion of the general version. |
24.10.2023 | Beginning of Chapter 3 (the sum-product phenomenon). Discussion of simple examples, including the Erdös "multiplication table problem". First proof of the sum-product theorem for integers (Erdös-Szemerédi). The argument of Elekes using the Szemerédi-Trotter Theorem. |
31.10.2023 | Sketch of the proof of the Szemerédi-Trotter inequality using the crossing inequality for planar graphs. Solymosi's proof of the sum-product inequality. Statement of the Bourgain-Katz-Tao Theorem. |
2.11.2023 | Proof of the Bourgain-Katz-Tao Theorem, following the original argument. Remarks on applications. |
7.11.2023 | Second proof of the Bourgain-Katz-Tao Theorem, using Breuillard's approach. |
14.11.2023 | Discussion of applications of the sum-product theorem: incidence bounds, Dvir's Theorem on Kakeya sets, exponential sums over multiplicative subgroups. |
16.11.2023 One hour lecture only |
Rough outline of the proof of the theorem of Bourgain, Glibichuk and Konyagin. |
21.11.2023 | General discussion of arithmetic progressions in subsets of abelian groups. Behrend's example of large sets without three term progressions. Presentation of the strategy of Roth's Theorem. |
28.11.2023 | Start of the proof of Roth's Theorem; comparison with the sumfree set situation. |
30.11.2023 | End of the proof of Roth's Theorem. Start of discussion of longer progressions. |
5.12.2023 | No lecture |
12.12.2023 | Definition of Gowers norms. Relation of the second Gowers norm with the Fourier transform. Proof that the Gowers norms are (semi)norms. |
14.12.2023 | Gowers norms control the behavior of functions along arithmetic progressions. Trichotomy for the existence of k-term progressions in terms of the size of a Gowers norm of the balanced characteristic function. Example of functions with large and small Gowers norms. |
14.12.2023 | Quick survey of the inverse theorem for Gowers norms. Going back to sumsets: discussion of the Freiman-Ruzsa Theorem, and outline of Ruzsa's proof. |
21.12.2023 | No lecture! |