## Topics in arithmetic combinatorics

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##### Authors

Sanders, Tom

##### Advisors

Gowers, William Timothy

##### Date

2007-10-23##### Awarding Institution

University of Cambridge

##### Author Affiliation

Department of Pure Mathematics and Mathematical Statistics

##### Qualification

Doctor of Philosophy (PhD)

##### Language

English

##### Type

Thesis

##### Metadata

Show full item record##### Citation

Sanders, T. (2007). Topics in arithmetic combinatorics (Doctoral thesis). https://doi.org/10.17863/CAM.16213

##### Description

E-thesis pagination differs from approved hard bound copy, Cambridge University Library classmark: PhD.30726

##### Abstract

This thesis is chiefly concerned with a classical conjecture of Littlewood's regarding the L^1-norm of the Fourier transform, and the closely related idempotent theorem. The vast majority of the results regarding these problems are, in some sense, qualitative or at the very least infinitary and it has become increasingly apparent that a quantitative state of affairs is desirable.
Broadly speaking, the first part of the thesis develops three new tools for tackling the problems above: We prove a new structural theorem for the spectrum of functions in A(G); we extend the notion of local Fourier analysis, pioneered by Bourgain, to a much more general structure, and localize Chang's classic structure theorem as well as our own spectral structure theorem; and we refine some aspects of Freiman's celebrated theorem regarding the structure of sets with small doubling. These tools lead to improvements in a number of existing additive results which we indicate, but for us the main purpose is in application to the analytic problems mentioned above.
The second part of the thesis discusses a natural version of Littlewood's problem for finite abelian groups. Here the situation varies wildly with the underlying group and we pay special attention first to the finite field case (where we use Chang's Theorem) and then to the case of residues modulo a prime where we require our new local structure theorem for A(G). We complete the consideration of Littlewood's problem for finite abelian groups by using the local version of Chang's Theorem we have developed. Finally we deploy the Freiman tools along with the extended Fourier analytic techniques to yield a fully quantitative version of the idempotent theorem.

##### Keywords

Fourier analysis, Freiman's theorem, Chang's theorem, Additive combinatorics, Littlewood's conjecture, Arithmetic combinatorics, Bourgain systems, Idempotent theorem, Local Fourier analysis, Induction on doubling, Discrete analysis