Combinatorics with a view towards number theory

Abstract

We study several problems in arithmetic combinatorics, loosely grouped around the themes of "combinatorial number theory" and "higher-order Fourier analysis". The problems related to the former theme are the Lonely Runner Problem (from Diophantine approximation), questions about minimal additive complements, and a conjecture of Graham about rearrangements of subsets of F_p^\times. The problems related to the latter theme concern quantitative bounds for some instances of the Bergelson--Leibman Theorem (about subsets of N^d avoiding polynomial progressions).