Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"
Image for the paper "The sum-product problem for integers with few prime factors"

Sum-product with few primes

Number theory

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.

The sum-product problem for integers with few prime factors

Submitted (2023)

I. Shkredov, B. Hanson, M. Rudnev, D. Zhelezov

It was asked by E. Szemerédi if, for a finite set AZA\subset \mathbb{Z}, one can improve estimates for maxA+A,AA\max{|A+A|,|A\cdot A|}, under the constraint that all integers involved have a bounded number of prime factors -- that is, each aAa\in A satisfies ω(a)k\omega(a)\leq k. In this paper, we answer Szemerédi's question in the affirmative by showing that this maximum is of order A5/3o(1)|A|^{5/3−o(1)} provided k(logA)1ϵk\leq(\log|A|)^{1−\epsilon} for some ϵ>0\epsilon>0. In fact, this will follow from an estimate for additive energy which is best possible up to factors of size Ao(1)|A|^{o(1)}.

Submitted (2023)

I. Shkredov, B. Hanson, M. Rudnev, D. Zhelezov