In particular, I like thinking about problems which start with a very simple object, with some simple global structural assumptions, and try to use that information to show that this object must have some more complex local structure. For example:

- How large can a set \(A\subset\{1,\ldots,N\}\) be before it is forced to contain a progression \(x,x+d,x+2d\), with \(d\neq 0\)?
- If a set \(A\subset \{1,\ldots,N\}\) has a small sum set, so \(\lvert A+A\rvert \ll \lvert A\rvert\), then is it forced to resemble an arithmetic progression?
- Given two graphs \(G\) and \(H\), when is it the case that any two-colouring of the edges of \(G\) must contain a monochromatic copy of \(H\)?

#### Papers

*Translation invariant equations and the method of Sanders*. Bulletin of the London Mathematical Society (2012), doi: 10.1112/blms/bds045.

Abstract | PDF | arxiv:1107.1110.

In which I extend the improvement of Roth's theorem on three term arithmetic progressions by Sanders to obtain similar results for the problem of locating non-trivial solutions to translation-invariant linear equations in many variables (e.g. \(x_1+x_2+x_3=3x_4\)) in both \(\mathbb{Z}/N\mathbb{Z}\) and \(\mathbb{F}_q[t]\).

(joint with T. G. F. Jones) *A sum-product theorem in function fields*. International Mathematics Research Notices (2013), doi: 10.1093/imrn/rnt125.

Abstract | PDF | arxiv:1211.5493.

In which we show that, for any finite set \(A\) which lives in either a rational function field \(\mathbb{F}_q(t)\) or a p-adic field \(\mathbb{Q}_p\), either the sum set \(A+A\) or the product set \(A\cdot A\) has cardinality at least \(\lvert A\rvert^{6/5-o(1)}\).

*A quantitative improvement for Roth's theorem on arithmetic progressions*. submitted.

arxiv:1405.5800.

In which I obtain an improved quantitative version of Roth's theorem, showing that if \(A\subset \{1,\ldots,N\}\) contains no non-trivial three-term arithmetic progressions then \(\lvert A\rvert \ll N(\log\log N)^4/\log N\). The method used is quite different to that used by Sanders, who earlier obtained a similar bound, and is based on the techniques used by Bateman and Katz in their work on the analogous problem over \(\mathbb{F}_3^n\).

(joint with A. Liebenau) *Ramsey equivalence of \(K_n\) and \(K_n+K_{n-1}\)*. submitted.

arxiv:1508.03866.

In which we show that for \(n\geq 4\) the graphs \(K_n\) and \(K_n+K_{n-1}\) are Ramsey equivalent; that is, if a graph \(G\) has the property that, given any red-blue colouring of the edges of \(G\), a monochromatic copy of \(K_n\) necessarily appears somewhere, then a monochromatic copy of \(K_n+K_{n-1}\) is also forced.

#### Thesis

My PhD thesis, completed in 2014 under the supervision of Professor Trevor Wooley, was titled "Quantitative topics in arithmetic combinatorics". A PDF copy can be found here.It includes, in particular, the main results of my first three papers (including some joint work with T. G. F. Jones). The results related to Roth's theorem are proved there in a more unified manner, however, and as a result several new technical corollaries are obtained.

There is also some otherwise-unpublished work on Freiman-type inverse theorems in polynomial rings.

#### Talks

- July 2015:
*Higher-order additive structure*at the BIRS Workshop "Combinatorics Meets Ergodic Theory" (video of this, and all the other talks at this workshop, are available here). - November 2014:
*On Roth's theorem on arithmetic progressions*for the Number Theory seminar, University of Oxford. - November 2014:
*Structure in large spectra*for the DIMAP seminar, University of Warwick. - June 2012:
*Arithmetic combinatorics in polynomial rings*at the "Analytic methods for Diophantine problems" conference, University of Gottingen. - August 2011:
*Polynomial rings in additive combinatorics*at the Paul Turan Memorial Conference in Mathematics, Budapest.