Arithmetic structure in sets of integers

This dissertation deals with four problems concerning arithmetic structures in densesets of integers. In Chapter 1 we give an exposition of the state-of-the-art techniquedue to Pintz, Steiger and Szemer edi which yields the best known upper bound onthe density of sets whose di erence set is square-free. Inspired by the well-knownfact 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 linearequations which guarantees the correct number of solutions in any uniform subset ofFnp. This joint work with Tim Gowers constitutes the core of this thesis and reliesheavily on recent progress in so-called 'quadratic Fourier analysis' pioneered by Gowers,Green and Tao. In particular, we use a structure theorem for bounded functionswhich provides a decomposition into a quadratically structured and a quadraticallyuniform part. We also present an alternative decomposition leading to improvedbounds for the main result, and discuss the connections with recent results in ergodictheory. Chapter 3 deals with improved upper and lower bounds on the minimumnumber 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 subsetof ZN. More precisely, we establish that, given a subset of size linear in N, the set ofits popular differences does not always contain the complete difference set of anotherlarge set.EThOS - Electronic Theses Online ServiceGBUnited Kingdo