Arithmetic combinatorics on Vinogradov systems

Finite field models in arithmetic combinatorics - ten years on

Math 8440 -arithmetic Combinatorics -spring 2011

A typical result in (additive) Ramsey theory takes the following form: if N (or {1, . . . , N} with N sufficiently large) is partitioned into finitely many classes, then at least one of these classes will contain contain a specific arithmetic structure (e.g. an arithmetic progression). The simplest example of such a result is the pigeonhole principle and one can view Ramsey theory as the study ...

Finite Field Models in Arithmetic Combinatorics

The study of many problems in additive combinatorics, such as Szemerédi’s theorem on arithmetic progressions, is made easier by first studying models for the problem in F p , for some fixed small prime p. We give a number of examples of finite field models of this type, which allows us to introduce some of the central ideas in additive combinatorics relatively cleanly. We also give an indicatio...

Combinatorics of Integer Partitions in Arithmetic Progression

The partitions of a positive integer n in which the parts are in arithmetic progression possess interesting combinatorial properties that distinguish them from other classes of partitions. We exhibit the properties by analyzing partitions with respect to a fixed length of the arithmetic progressions. We also address an open question concerning the number of integers k for which there is a k-par...

From Harmonic Analysis to Arithmetic Combinatorics

We will describe a certain line of research connecting classical harmonic analysis to PDE regularity estimates, an old question in Euclidean geometry, a variety of deep combinatorial problems, recent advances in analytic number theory, and more. Traditionally, restriction theory is a part of classical Fourier analysis that investigates the relationship between geometric and Fourier-analytic pro...