Syllabus

 Course contents of MAL511: Additive Number Theory

Sumsets: Introduction to sumsets - Direct and inverse problems for sumsets. Sumsets in torsion-free abelian groups: Freiman homomorphisms and Freiman isomorphisms, basic lower bounds and inverse theorems for sumsets.

Sizes of sumsets: Doubling constant - Ruzsa distance - additive energy -  Ruzsa’s covering lemma, Green-Ruzsa covering lemma - Sidon sets  - Freiman's 3k-4 theorem - sum-product problems.

Sumsets in groups: Cauchy-Davenport theorem - Pollard's theorem - Erdős-Ginzburg-Ziv theorem - Chevalley-Warning theorem - Vosper's theorem - Freiman-Vosper theorem - Kemperman’s theorem - Kneser's addition theorem and its applications - Rectification principles.

The polynomial method: Alon’s combinatorial Nullstellensatz - restricted sumsets and the Erdős-Heilbronn conjecture - Dias da Silva-Hamidoune theorem - Snevily’s conjecture - Kemnitz’s conjecture.

Sumsets in higher dimensional Euclidean space. Geometry of numbers: Lattices and determinants - Minkowski's first theorem and Minkowski's second theorem. 

Structure of sets with small sumsets: Plünnecke-Ruzsa theorem - review of Fourier analysis on groups - Bohr  sets in sumsets - Bogolyubov's Lemma -  generalized arithmetic progression in Bohr sets - Freiman-Ruzsa Theorem.