Topics in arithmetic combinatorics

163 Pages·2011·0.68 MB·English

Topics in arithmetic combinatorics
Tom Sanders
Department of Pure Mathematics
and Mathematical Statistics
A dissertation submitted for
the degree of Doctor of Philosophy
at the University of Cambridge.

To Mum & Dad

This dissertation is the result of my own work and includes nothing which
is the outcome of work done in collaboration except where specifically indi-
cated in the text.

Abstract
This thesis is chiefly concerned with a classical conjecture of Littlewood’s re-
garding the L1-norm of the Fourier transform, and the closely related idem-
potent theorem. The vast majority of the results regarding these problems
are, insomesense, qualitativeorattheveryleastinfinitaryandithasbecome
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 anal-
ysis, pioneered by Bourgain, to a much more general structure, and localize
Chang’s classic structure theorem as well as our own spectral structure the-
orem; and we refine some aspects of Fre˘ıman’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 Fre˘ıman tools along with the extended Fourier analytic techniques
to yield a fully quantitative version of the idempotent theorem.

Acknowledgements
I should like to thank my supervisor Tim Gowers for his support and en-
couragement and Ben Green for functionally acting as a second supervisor
despite no call of duty to do so. I should also like to thank the range of
mathematicians with whom I have had many useful conversations over these
last few years.
Tom Sanders
Cambridge
8th December 2006