I started reading a very recent paper due to J. Bourgain and A. Kontorovich titled “On Zaremba’s conjecture”, in the Annals of Mathematics . The paper was intriguing to me because the principal technique in the paper is the circle method, but the set-up involved an affine sieve flavor. Of course, Bourgain is one of the principal authors behind the affine sieve (the others being Gamburd and Sarnak), so this should not be unexpected. This paper and the application of the circle method is interesting to me because it is related to a problem I am working on connecting Sidon sets and additive bases of the natural numbers.
A major component behind the affine sieve is the expander graph. These are graphs which are relatively sparse, but very robust. They have the critical property that a random walk on the graph is very rapidly mixing, so a random walk approximates uniform distribution. Their application in number theory is based on the observation that for a finitely generated group, the Cayley graph of the group on a (symmetric) set of generators forms an expander. This non-trivial connection allows one to subsume many famous results in number theory, including Chen’s theorem, in terms of a single theory. I aim to do some applied mathematics research involving expander graphs in the near future as well.