I'm interested in number theory, combinatorics, and especially the problems which arise from the intersection of the two fields, living in an area known as arithmetic combinatorics.

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:

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 \(H\)?

In this expository note, we give a concise and accessible introduction to the real-analytic determinant method for counting integral points on algebraic curves, based on the classic 1989 paper of Bombieri-Pila.

We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application we prove near-optimal upper bounds for the average of the refined singular series in the Hardy-Littlewood conjectures concerning the number of prime $k$-tuples for $k$ odd. The main new ingredient is a near-optimal upper bound for the number of solutions to $\sum_{1\leq i\leq k}\frac{a_i}{q_i}\in\mathbb{Z}$ when $k$ is odd, with $(a_i,q_i)=1$ and restrictions on the size of the numerators and denominators, that is of independent interest.

(joint with O. Sisask)

An improvement to the Kelley-Meka bounds on three-term arithmetic progressions

Submitted.

In a recent breakthrough Kelley and Meka proved a quasipolynomial upper bound for the density of sets of integers without non-trivial three-term arithmetic progressions. We present a simple modification to their method that strengthens their conclusion, in particular proving that if $A\subseteq \{1,\ldots,N\}$ has no non-trivial three-term arithmetic progressions then $\lvert A\rvert \leq \exp(-c(\log N)^{1/9})N$ for some $c > 0$.

Unit fractions with shifted prime denominators

Proceedings of the Royal Society of Edinburgh Section A:Mathematics

We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p \textrm{ prime}\}$. The same conclusion holds for the set $\{ p-h : p \textrm{ prime}\}$ for any $h\in\mathbb{Z}\backslash \{0\}$, provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.

(joint with O. Sisask)

The Kelley-Meka bounds for sets free of three-term arithmetic progressions

Essential Number Theory, Vol. 2 (2023), No. 1, 15-44.

We give a self-contained exposition of the recent remarkable result of Kelley and Meka: if $A\subseteq \{1,\ldots,N\}$ has no non-trivial three-term arithmetic progressions then $\lvert A\rvert \leq \exp(-c(\log N)^{1/12})N$ for some constant $c > 0$. Although our proof is identical to that of Kelley and Meka in all of the main ideas, we also incorporate some minor simplifications relating tog Bohr sets. This eases some of the technical difficulties tackled by Kelley and Meka and widens the scope of their method. As a consequence, we improve the lower bounds for finding long arithmetic progressions in $A+A+A$, where $A\subseteq \{1,\ldots,N\}$.

(joint with C. Elsholtz)

Egyptian Fractions

Nieuw Archief voor Wiskunde, December 2022, 237-245.

Any rational number can be written as the sum of distinct unit fractions. In this survey paper we review some of the many interesting questions concerning such 'Egyptian fraction' decompositions, and recent progress concerning them.

We prove that any set $A\subset \mathbb{N}$ of positive upper density contains a finite $S\subset A$
such that $\sum_{n\in S}\frac{1}{n}=1$, answering a question of Erdős and Graham.

(joint with J. Maynard)

A new upper bound for sets with no square differences

Compositio Mathematica, Vol. 158 (2022), Issue 8, 1777-1798.

We show that if $A\subset \{1,\ldots,N\}$ has no solutions to $a-b=n^2$ with $a,b\in A$ and $n\geq 1$ then
\[ \lvert A\rvert \ll \frac{N}{(\log N)^{c\log\log\log N}}\]
for some absolute constant $c>0$. This improves upon a result of Pintz-Steiger-Szemerédi.

Quantitative inverse theory of Gowers uniformity norms

Bourbaki seminar (originally planned for March 2020, but rescheduled due to the pandemic).

(This text is a survey written for the Bourbaki seminar on the work of F. Manners.)

Gowers uniformity norms are the central objects of higher order Fourier analysis, one of the cornerstones of additive combinatorics, and play an important role in both Gowers' proof of Szemerédi's theorem and the Green-Tao theorem. The inverse theorem states that if a function has a large uniformity norm, which is a robust combinatorial measure of structure, then it must correlate with a nilsequence, which is a highly structured algebraic object. This was proved in a qualitative sense by Green, Tao, and Ziegler, but with a proof that was incapable of providing reasonable bounds. In 2018 Manners achieved a breakthrough by giving a new proof of the inverse theorem. Not only does this new proof contain a wealth of new insights but it also, for the first time, provides quantitative bounds, that are at worst only doubly exponential. This talk will give a high-level overview of what the inverse theorem says, why it is important, and the new proof of Manners.

Gowers uniformity norms are the central objects of higher order Fourier analysis, one of the cornerstones of additive combinatorics, and play an important role in both Gowers' proof of Szemerédi's theorem and the Green-Tao theorem. The inverse theorem states that if a function has a large uniformity norm, which is a robust combinatorial measure of structure, then it must correlate with a nilsequence, which is a highly structured algebraic object. This was proved in a qualitative sense by Green, Tao, and Ziegler, but with a proof that was incapable of providing reasonable bounds. In 2018 Manners achieved a breakthrough by giving a new proof of the inverse theorem. Not only does this new proof contain a wealth of new insights but it also, for the first time, provides quantitative bounds, that are at worst only doubly exponential. This talk will give a high-level overview of what the inverse theorem says, why it is important, and the new proof of Manners.

(joint with O. Sisask)

Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions

Submitted.

We show that if $A\subset \{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions then $\lvert A\rvert \ll N/(\log N)^{1+c}$ for some absolute constant $c>0$. In particular, this proves the first non-trivial case of a conjecture of Erdős on arithmetic progressions.

(joint with O. Sisask)

Logarithmic bounds for Roth's theorem via almost-periodicity

Discrete Analysis 2019:4.

We give a new proof of logarithmic bounds for Roth's theorem on arithmetic progressions, namely that if $A\subset \{1,2,\ldots,N\}$ is free of three-term progressions, then $\lvert A\rvert \leq N/(\log N)^{1-o(1)}$. Unlike previous proofs, this is almost entirely done in physical space using almost-periodicity.

(joint with A. Walker)

GCD sums and sum-product estimates

Israel Journal of Mathematics (2019).

In this note we prove a new estimate on so-called GCD sums (also called Gál sums), which, for certain coefficients, improves significantly over the general bound due to de la Bretêche and Tenenbaum. We use our estimate to prove new results on the equidistribution of sequences modulo 1, improving over a result of Aistleitner, Larcher, and Lewko on how the metric poissonian property relates to the notion of additive energy. In particular, we show that arbitrary subsets of the squares are metric poissonian.

(joint with S. Chow, A. Gafni, and A. Walker)

Additive energy and the metric Poissonian property

Mathematika 64 (2018) 679-700.

Roughly, a set $A$ of natural numbers is metric Poissonian if the dilates of differences $\alpha(a-b)$ for $\alpha \in (0,1)$ and $a,b\in A$ are on average well-distributed. We show that if $A$ has small additive energy, i.e. relatively few solutions to $a-b=c-d$, and is relatively dense, then $A$ is metric Poissonian.

(joint with A. Liebenau)

Ramsey equivalence of \(K_n\) and \(K_n+K_{n-1}\)

The Electronic Journal of Combinatorics 25 (2018) #P3.4.

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.

A quantitative improvement for Roth's theorem on arithmetic progressions

Journal of the London Mathematical Society (2016).

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 T. G. F. Jones)

A sum-product theorem in function fields

International Mathematics Research Notices (2013).

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 \(\lvert A\rvert^{6/5-o(1)}\).

Translation invariant equations and the method of Sanders

Bulletin of the London Mathematical Society (2012).

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 \(\mathbb{F}_q[t]\).

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.