On arithmetic structures in dense sets of integers

Abstract

This dissertation deals with four problems concerning arithmetic structures in dense sets of integers. In Chapter 1 we give an exposition of the state-of-the-art technique due to Pintz, Steiger and Szemer edi which yields the best known upper bound on the density of sets whose di erence set is square-free. Inspired by the well-known fact that Fourier analysis is not su cient to detect progressions of length 4 or more, we determine in Chapter 2 a necessary and sufficient condition on a system of linear equations which guarantees the correct number of solutions in any uniform subset of Fnp. This joint work with Tim Gowers constitutes the core of this thesis and relies heavily on recent progress in so-called "quadratic Fourier analysis" pioneered by Gowers, Green and Tao. In particular, we use a structure theorem for bounded functions which provides a decomposition into a quadratically structured and a quadratically uniform part. We also present an alternative decomposition leading to improved bounds for the main result, and discuss the connections with recent results in ergodic theory. Chapter 3 deals with improved upper and lower bounds on the minimum number of monochromatic 4-term progressions in any two-colouring of ZN. Finally, in Chapter 4 we investigate the structure of the set of popular di erences of a subset of ZN. More precisely, we establish that, given a subset of size linear in N, the set of its popular differences does not always contain the complete difference set of another large set.