Misha Rudnev: The sum-product problem for integers with few prime factors

It was asked by Szemerédi if the known sum-product estimates can be improved for a set of N integers under the constraint that each integer has a small number of prime factors. We prove, if the maximum number of prime factors for each integer is sub-logarithmic in N, the sum-product exponent 5/3-o(1). 

This becomes a corollary of an additive energy versus the product set cardinality estimate, which turns out to be the best possible. It is based on a scheme of Burkholder-Gundy-Davis martingale square function inequalities in p-adic scales, followed by an application of a variant of the Schmidt subspace theorem.

The zoom link to the talk is:

https://us06web.zoom.us/j/97594629945?pwd=MmFNaVk4a1FhdjEvc2RRdGdod0FpZz09 .