Brief Summaries of Ben's Expository Notes
An essay on the Kakeya Problem, Montgomery's Large Values Conjecture and the Density Hypothesis on zeros of 
. This essay explores some fascinating links between geometric measure theory, combinatorics and analytic number theory.
It is now nearly 2 years since I wrote it however, and upon reading it again I was struck by how well I managed to obscure the geometric motivation behind many of the arguments. In addition the essay is now out of date in that various combinations of Bourgain, Tao, Katz and Laba have improved the best known bounds. I suggest reading the various expository articles by Terence Tao, though none of these cover the connection with Montgomery's conjecture.
An Exposition of Jean Bourgain's paper "On Triples in Arithmetic Progression", in which it is shown that a subset of {1,...,N} of density (loglogN/logN)1/2 contains a triple in arithmetic progression. A technically dazzling paper, but not easy reading - I spent a long time writing this.
A slightly novel slant on a paper of Szemer\'edi in which it is shown that there is a (small) constant c such that any subset of {1,...,N} of density (logN)-c contains a triple in arithmetic progression. Although the bound is inferior to that of Bourgain (see above) this paper is a great deal easier to understand, and the result suffices for a certain application in the Bourgain/Katz/Tao work on the Hausdorff dimension of Kakeya sets. I have recently added some opinions about where the strengths and weaknesses of this approach lie.
An exposition of another paper of Bourgain, this time concerning the location of long APs in sets of form A + B with A and B dense subsets of {1,...,N}. Roughly speaking the main result here is that if |A| = aN and |B| = bN, where a and b are to be thought of as fixed, then A + B conatins a progression of size about e(log N)1/3. The proof uses a result of Rudin from the theory of thin sets in harmonic analysis, and we prove this here in full.
This is a bit of a ramble through the basic theory of elliptic curves and modular forms, culminating in a (virtually) calculation-free proof that Ramanujan's number e
\sqrt{163} is within 10-12 of an integer. I wrote this before I decided not to become an algebraic number theorist, but have received some favourable comments on it. Hence I've decided there's no harm in making it available.
B.J. Green
Last modified: Sat Feb 17 21:01:45 EST 2001