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I am interested in many questions in additive combinatorics and number theory. My main research topics are: Roth's theorem and its extensions; applications of analytic ideas to combinatorial problems; higher-order Fourier analysis; extremal structures in additive combinatorics and the structure of structure-avoiding sets; computational aspects of additive number theory. I am particularly interested in ways in which one can isolate the important aspects of some data and represent these using a small number of parameters.
Papers and articles
Roth's theorem for four variables and additive structures in sums of sparse sets.
With Tomasz Schoen.
Convergence results for systems of linear forms on cyclic groups, and periodic nilsequences.
With Pablo Candela. To appear in SIAM J. Discrete Math.
Arithmetic progressions in sumsets and Lp-almost-periodicity.
With Ernie Croot and Izabella Laba. Combin. Probab. Comput. 22 (3) (2013), 351–365.
A removal lemma for linear configurations in subsets of the circle.
With Pablo Candela. Proc. Edinburgh Math. Soc. 56 (3) (2013), 657–666.
On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime.
With Pablo Candela. Acta Math. Hungar. 132 (3) (2011), 223–243.
What is the largest density of a subset of Z/pZ that does not contain a solution to either of the equations x1 + x2 = x3, x1 + 2x2 = 4x3 + 5x4? This paper shows that if p is large then the answer to such questions is essentially given by an analogous quantity in R/Z, answering a Z/pZ-version of a question of Ruzsa about subsets of [N].
A probabilistic technique for finding almost-periods of convolutions.
With Ernie Croot. Geom. Funct. Anal. 20 (2010), 1367–1396.
On the maximal number of three-term arithmetic progressions in subsets of Z/pZ.
With Ben Green. Bull. London Math. Soc. 40 (2008), 945–955.
A new proof of Roth's theorem on arithmetic progressions.
With Ernie Croot. Proc. Amer. Math. Soc. 137 (2009), 805–809.
Freiman isomorphisms between characters and linear limits of groups.
We prove that the minimum number of 3APs in a subset of Z/pZ of density delta, divided by p2, is the same as the minimum amount of 3APs in a subset of the R/Z of density delta, up to o(1) errors as p tends to infinity. In fact, our results are rather more general than this, dealing with the general question of moving between linear equation counts over any compact abelian groups. E-mail me if interested!
(still needs polishing)
Bourgain's proof of the existence of long arithmetic progressions in A+B.
These are some notes I wrote while trying to understand Bourgain's proof of the existence of long arithmetic progressions in sumsets A+B. I found it easiest to think of Bourgain's work as establishing an Lp-almost-periodicity result for convolutions of functions.
A family of large density, large diameter sum-free sets in Z/pZ.
Work of Deshouillers, Freiman and Lev has shown that large sum-free subsets of Z/pZ are necessarily somewhat structured, in the sense that they have a dilate contained in a short interval. In particular, this is known to hold for sets of density at least 0.318. In this note we construct a family of sum-free sets of density 0.25 that do not have a dilate contained in a short interval.
An additive combinatorial take on Zeta constants.
This is a short note demonstrating how one may interpret the values of Zeta(2), Zeta(4) and related sums in an additive combinatorial fashion. The basic idea is that one can view these values as representing the number of solutions to some linear equation in a simple subset of Z/pZ. What about Zeta(3)? Read the note and try it yourself!