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.
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.
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.
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