More on the sum-product problem for integers with few prime factors
Thomas F. Bloom
TL;DR
This work studies the sum-product phenomenon for finite sets of integers with few prime factors. It strengthens prior results by proving that, when every element of A has at most k prime factors, max(|A+A|,|AA|) grows as |A|^{17/10-o(1)}, surpassing the Balog–Wooley barrier. It also shows that for every m≥2, max(|mA|,|A^m|) ≥ |A|^{(2/3)m+1/3-o(1)} under the same hypothesis. The authors achieve this via a streamlined approach that reduces A to a large subset efficiently contained in a bounded-rank multiplicative group, leverages S-unit bounds and Amoroso–Viada-type results, and exploits higher additive energies E_{2m}(A) together with Hölder’s inequality. These methods yield improved sumset lower bounds and sharp growth for iterated sums/products, advancing understanding of sum-product phenomena in restricted prime-factor regimes.
Abstract
We show that if $A\subset \mathbb{Z}$ is a finite set of integers in which every integer is divisible by $O(1)$ many primes then \[\max(\lvert A+A\rvert,\lvert AA\rvert) \geq \lvert A\rvert^{17/10-o(1)}\] and, for any $m\geq 2$, \[\max(\lvert mA\rvert, \lvert A^m\rvert) \geq \lvert A\rvert^{\frac{2}{3}m+\frac{1}{3}-o(1)}.\]