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:
- 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 (in reverse chronological order)
(joint with W. Sawin, C. Schildkraut, and D. Zhelezov)
The sum-product conjecture is false for real numbers
arxiv:2605.28781
We disprove the sum-product conjecture for real numbers by constructing arbitrarily large $A\subset \mathbb{R}$ (whose elements are algebraic integers in a number field of degree $\asymp \log\lvert A\rvert$) such that \[\max(\lvert A+A\rvert ,\lvert AA\rvert)\leq \lvert A\rvert^{2-c}\] where $c>0$ is an absolute constant.
We also disprove the many sums and products conjecture by constructing, for any $k\geq 3$, arbitrarily large $A\subset \mathbb{R}$ such that \[\max(\lvert kA\rvert,\lvert A^{(k)}\rvert)\leq \lvert A\rvert^{C\frac{\log k}{\log\log k}}\] for some constant $C>0$. We obtain similar constructions for $p$-adics, finite fields, and function fields in positive characteristic, and also obtain new lower bounds for the number of solutions to linear equations in a multiplicative group and the number of solutions to the unit equation in sufficiently many variables.
(joint with N. Alon, W. T. Gowers, D. Litt, W. Sawin, A. Shankar, J. Tsimerman, V. Wang, and M. M. Wood)
Remarks on the disproof of the unit distance conjecture
arxiv:2605.20695
We present a short, digested, human-verified version of the recent OpenAI-generated counterexample to the Erdős unit distance conjecture, and a sequence of reflections on it. The argument relies crucially on ideas that may, at least in retrospect, be attributed to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna.
The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if $A\subset \mathbb{Z}$ is a finite set of integers with $\lvert A\rvert=N$ then $\| \widehat{1_A}\|_1\geq c\log N$ for some absolute constant $c > 0$. We explore what structure $A$ must have if $\| \widehat{1_A}\|_1\leq K\log N$ for some constant $K$. Under such an assumption we prove, for instance, that $A$ contains a subset $A'\subseteq A$ with $\lvert A\rvert \geq N^{0.99}$ such that $\lvert A'+A'\rvert \ll K^{O(1)}\lvert A'\rvert$. As a consequence, for any $k\geq 3$, if $N$ is sufficiently large depending on $k$ and $K$, then $A$ must contain an arithmetic progression of length $k$. A byproduct of our analysis is a (slightly) improved bound for the constant $c$.
(joint with R. Agrawal and G. Petridis)
More on the sum-product problem for integers with few prime factors
arxiv:2512.04931
We show that if $A\subset \mathbb{Z}$ is a finite set of integers in which every integer is divisible by $O(1)$ many primes then \[\max(\lvert A+A\rvert,\lvert AA\rvert) \geq \lvert A\rvert^{12/7-o(1)}\] and, for any $m\geq 2$, \[\max(\lvert mA\rvert, \lvert A^{(m)}\rvert) \geq \lvert A\rvert^{\frac{2}{3}m+\frac{1}{3}-o(1)}.\] Finally, we show that if $A\subset \mathbb{Q}$ is a finite set of rationals in which the numerator and denominator of every $x\in A$ is divisible by $O(1)$ many primes then $\lvert A+AA\rvert \geq \lvert A\rvert^{2-o(1)}$.
(joint with E. Croot)
Integers with small digits in multiple bases
arxiv:2509.02835
We show that, for any $r\geq 1$, if $g_1,\ldots,g_r$ are distinct coprime integers, sufficiently large depending only on $r$, then for any $\epsilon>0$ there are infinitely many integers $n$ such that all but $\epsilon \log n$ of the digits of $n$ are $\leq g_i/2$ in base $g_i$ for all $1\leq i\leq r$. In other words, for any fixed large bases, there are infinitely many $n$ such that almost all of the digits of $n$ are small in all bases simultaneously. This is both a quantitative and qualitative improvement over previous work of Croot, Mousavi, and Schmidt. As a consequence, we obtain a weak answer to a conjecture of Graham concerning divisibility of $\binom{2n}{n}$.
(joint with J. Fuhrer and O. Roche-Newton)
Additive structure in convex sets
Combinatorics 46 (2026), no. 1, Paper No. 7, 16pp.
Journal copy | arxiv:2509.01568
This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set $A$ containing $\Omega(\lvert A\rvert^{3/2})$ three-term arithmetic progressions.
A polynomial Freiman-Ruzsa inverse theorem for function fields
Discrete Anal. 2025, Paper No. 24, 11 pp.
arxiv:2501.11580
Using the recent proof of the polynomial Freiman-Ruzsa conjecture over $\mathbb{F}_p^n$ by Gowers, Green, Manners, and Tao, we prove a version of the polynomial Freiman-Ruzsa conjecture over function fields. In particular, we prove that if $A\subset\mathbb{F}_p[t]$ satisfies $\lvert A+tA\rvert\leq K\lvert A\rvert$ then $A$ is efficiently covered by at most $K^{O(1)}$ translates of a generalised arithmetic progression of rank $O(\log K)$ and size at most $K^{O(1)}\lvert A\rvert$.
As an application we give an optimal lower bound for the size of $A+\xi A$ where $A\subset\mathbb{F}_p((1/t))$ is a finite set and $\xi\in \mathbb{F}_p((1/t))$ is transcendental over $\mathbb{F}_p[t]$.
We prove new quantitative bounds on the additive structure of sets obeying an $L^3$ 'control' assumption, which arises naturally in several questions within additive combinatorics. This has a number of applications - in particular we improve the known bounds for the sum-product problem, the Balog-Szemerédi-Gowers theorem, and the additive growth of convex sets.
(joint with J. D. Lichtman)
The Bombieri-Pila determinant method
Essent. Number Theory 4 (2025), no. 2, 327–348.
Journal copy | arxiv:2312.12890
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.
(joint with V. Kuperberg)
Odd moments and adding fractions
Proc. Lond. Math. Soc. (3) 131 (2025), no. 1, Paper No. e70068, 38 pp.
Journal copy | arxiv:2312.09021
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.
An improvement to the Kelley-Meka bounds on three-term arithmetic progressions
arxiv:2309.02353
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
Journal copy | arxiv:2305.02689
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.
The Kelley-Meka bounds for sets free of three-term arithmetic progressions
Essential Number Theory, Vol. 2 (2023), No. 1, 15-44.
Journal copy |
arxiv:2302.07211
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 $A\subseteq \{1,\ldots,N\}$.
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.
(with an appendix joint with B. Mehta)
On a density conjecture about unit fractions
J. Eur. Math. Soc. (JEMS) 27 (2025), no. 11, 4563–4589.
Journal copy | arxiv:2112.03726
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.
Journal copy |
arxiv:2011.13266
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).
arxiv:2009.01774
(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.
Breaking the logarithmic barrier in Roth's theorem on arithmetic progressions
arxiv:2007.03528
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.
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.
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.
Journal copy | arxiv:1709.02634
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.
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).
Journal copy | arxiv:1405.5800
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).
Journal copy | arxiv:1211.5493
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).
Journal copy | arxiv:1107.1110
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 \(\mathbb{F}_q[t]\).
Thesis
My PhD thesis, completed in 2014 under the supervision of Professor Trevor Wooley, is 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.