In new work with Misha Rudnev, we prove a stronger bound on the sum-product problem, showing that
max(|A + A|, |AA|) ≥ |A|4/3 + 2/1167 − o(1)
for any finite set A of real numbers. This builds upon the work of Solymosi, Konyagin and Shkredov, although our paper is self-contained. I will give an overview of the arguments, both old and new, and describe some consequences of the new arguments.
How many primes are there which are less than x and congruent to a modulo q? This is one of the most important questions in analytic number theory, but also one of the hardest - our current knowledge is limited, and any direct improvements require solving exceptionally difficult questions to do with exceptional zeros and the Generalized Riemann Hypothesis! If we ask for 'averaged' results then we can do better, and powerful work of Bombieri and Vinogradov gives good answers for q less than the square-root of x. For many applications this is as good as the Generalized Riemann Hypothesis itself! Going beyond this 'square-root' barrier is a notorious problem which has been achieved only in special situations, perhaps most notably this was the key component in the work of Zhang on bounded gaps between primes. I'll talk about recent work going beyond this barrier in some new situations. This relies on fun connections between algebraic geometry, spectral theory of automorphic forms, Fourier analysis and classical prime number theory. The talk is intended for a general audience.
Given a sequence a1, a2, . . . of integers, one can form the sequence |a1 - a2|, |a2, a3|, . . .. Gilbreath's conjecture is that if you start with the sequence of the primes and iterate this consecutive differencing procedure, then the first term of every sequence (besides the initial one) is a 1. We prove the conclusion of Gilbreath's conjecture for a suitably random initial sequence instead of the primes.
The Kakeya conjectures of harmonic analysis claim that congruent tubes that point in different directions rarely meet. In this talk we discuss the resolution of an analogous problem in a discrete setting (where the tubes are replaced by lines), and provide some structural information on quasi-extremal configurations.
In a very interesting paper, Gowers and Long discussed binary operations * on finite sets which are somewhat associative in the sense that x * (y * z) = (x * y) * z for 1 percent (say) of all triples (x,y,z). They presented an example of such an operation which, they conjectured, is not closely related to any genuine group operation. I will discuss a proof of their conjecture, which uses a number of tools from (nonabelian) additive combinatorics.
In their seminal paper Erdős and Szemerédi formulated conjectures on the size of sum set and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph, when we consider sums and products of some pairs only. With Noga Alon and Imre Ruzsa we showed that this strong form of the Erdős-Szemerédi conjecture does not hold. In this talk I will list some related problems and recent results.
We improve the upper bound for diagonal Ramsey numbers to R(k+1,k+1) ≤ exp(-c(log k)2)(2k)!/(k!)2 for k ≥ 3. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal 'effective quasirandomness' results about convergence of graphs. This optimality represents a natural barrier to improvement.
Rephrasing the celebrated Freiman lemma in additive combinatorics, one can show that a finite set in ℤd containing a K-dimensional simplex has additive doubling at least ~K. We will discuss a novel framework for studying how such induced doubling can be inherited from a more general class of multi-dimensional subsets. It turns out that subsets of so-called quasi-cubes induce large doubling no matter the dimension of the ambient set. Time permitting, we will discuss how it allows to deduce a structural theorem for sets with polynomially large additive doubling and an application to the ”few products, many sums” problem of Bourgain and Chang.
I sketch a proof of a new bound in Roth's theorem on arithmetic progressions: if A ⊆ {1,...,N} does not contain any non-trivial three-term arithmetic progression then |A| ≪ (log log N)^3+o(1)N/log N.
Green proved the following strengthening of Roth's theorem: for every positive ϵ, there is some n(ϵ) such that for every N ≥ n(ϵ) and A ⊂ [N] with |A| = αN, there is some nonzero d such that A contains at least (α3 − ϵ)N three-term arithmetic progressions with common difference d (i.e., a popular common difference with frequently at least roughly the random bound). I'll discuss some extensions and generalizations of this result:
• How large does n(ϵ) have to be for the result to hold? (It turns out that a tower-type bound is necessary)
• Besides 3-term arithmetic progressions, is there a similar result for other patterns?
• What about patterns in higher-dimensional patterns?
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