Among topics that will be covered in the class are the following: discrete Fourier analysis over finite abelian groups; arithmetic progressions in the integers and finite abelian groups (Szemerédi's theorem); Sidon sets; sum-free sets in the integers and finite abelian groups; sum-product estimates (the Erdős–Szemerédi conjecture); the Littlewood–Offord problem; the structure of sets with a small sumset

There will be no final exam. Instead, the grade will be given based on solutions to homework assignments/exercises.

The course will be taught in English.

March 13

Van der Waerden's theorem; the Erdős–Turán conjecture / Szemerédi's theorem (without proof); characters of a finite abelian group and the Pontryagin dual; Fourier transform over finite abelian groups; Parseval's identity / Plancherel's formula

March 20

Fourier transforms of subsets of finite abelian groups; Roth's theorem in ℤ

_{3}^{m} (Meshulam's theorem) with proof; proof of Roth's theorem in {1, ..., n}

March 27

Progression of upper bounds in Roth's theorem; Freiman isomorphisms; Behrend's construction of large subsets of {1, ..., n} with no 3-term APs; the triangle removal lemma; derivation of Roth's theorem from the triangle removal lemma; the "corners" theorem in [n]

^{2} and derivation of Roth's theorem from it; the K

_{r+1}^{(r)} removal lemma; the "corners" theorem in [n]

^{d}
April 3

The notion of ε-regularity; Szemerédi's regularity lemma; the triangle counting lemma; proof of the triangle removal lemma; proof of Szemerédi's regularity lemma

April 24

Proof of Szmerédi's regularity lemma (continued); the upper bound on the size of 3AP-free sets in (ℤ

_{q})

^{m} due to Ellenberg–Gijswijt, after Croot–Lev–Pach; a proof of Hoeffding's inequality (large deviations of sums of independent bounded random variables)

May 8

Cauchy–Davenport theorem; Kneser's theorem; sum-free sets; largest sum-free subsets of {1, ..., n}; every n-element set of integers contains a sum-free subset with (n+1)/3 elements; the theorem of Eberhard, Green, and Manners (without proof)

May 15

Infinite sum-free sets either contain only odd numbers or have upper density at most 2/5 (Łuczak); Sidon sets in {1, ..., n}: the upper bound of Erdős and Turán and the construction of Singer / Ruzsa; B

_{h}-sets in {1, ..., n}: the trivial upper bound and the construction of Bose and Chowla; the sum-product conjecture of Erdős and Szemerédi; the upper bound of Erdős and Szemerédi

May 22

Progression of lower bounds in the sum-product conjecture; the argument of Elekes; the Szemerédi–Trotter incidence theorem; the crossing lemma of Ajtai–Chvátal–Newborn–Szemerédi; Solymosi's bound; the Littlewood–Offord problem; Sperner's theorem and Erdős' bound

May 29

Halász' theorem; singularity of random Bernoulli matrices

June 5

An analogue of the Cauchy–Davenport theorem in ℝ/ℤ; Frieman's 3k-4 theorem; Plünnecke graphs

June 12

Plünnecke graphs: addition graphs, independent addition graphs, product graphs, inverse graphs; magnification ratios; multiplicativity of magnification ratios; Plünnecke's theorem; estimates for the growth of iterated sumsets in abelian groups

June 19

Geometry of numbers: lattices, fundamental parallelepipeds, determinants; Blichfeldt's lemma; Minkowski's first and second theorem

June 26

Generalised arithmetic progressions (GAPs); Bohr neighbourhoods contain large GAPs; Bogolyubov's theorem; Freiman's inverse theorem