Our papers are the official record of our discoveries. They allow others to build on and apply our work. Each paper is the result of many months of research, so we make a special effort to make them clear, beautiful and inspirational, and publish them in leading journals.
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Number theory
IS
Canadian Journal of MathematicsMultiplicativity of sets
Expanding the known multiplicative properties of large difference sets yields a new, quantitative proof on the structure of product sets.
Combinatorics
VFIS
Journal of Algebraic CombinatoricsThe popularity gap
A finite nonempty subset A of a cyclic group, with small enough |A–A|, contains a nonzero element with at least (2+o(1))|A|²/|A–A| representations as a difference of two elements.
Combinatorics
IS
SubmittedRepresentation for sum-product
A new way to estimate indices via representation theory reveals links to the sum-product phenomena and Zaremba’s conjecture in number theory.
Combinatorics
IS
Finite Fields and Their ApplicationsUngrouped machines
A new connection between continued fractions and the Bourgain–Gamburd machine reveals a girth-free variant of this widely-celebrated theorem.
Number theory
Sum-product with few primes
For a finite set of integers with few prime factors, improving the lower bound on its sum and product sets affirms the Erdös-Szemerédi conjecture.
Number theory
Higher energies
Generalising the recent Kelley–Meka result on sets avoiding arithmetic progressions of length three leads to developments in the theory of the higher energies.
Number theory
Random Chowla conjecture
The distribution of partial sums of a Steinhaus random multiplicative function, of polynomials in a given form, converges to the standard complex Gaussian.
Number theory
Bounding Zaremba’s conjecture
Using methods related to the Bourgain–Gamburd machine refines the previous bound on Zaremba’s conjecture in the theory of continued fractions.
Combinatorics
Set additivity and growth
The additive dimension of a set, which is the size of a maximal dissociated subset, is closely connected to the rapid growth of higher sumsets.
Number theory
Energy bounds for roots
Bounds for additive energies of modular roots can be generalised and improved with tools from additive combinatorics and algebraic number theory.