Asymmetric estimates and the sum-product problems

TL;DR

This paper introduces new asymmetric estimates related to collinear triples and algebraic solutions, leading to improved bounds in difference-product/division problems and a refined Balog-Wooley decomposition for finite real sets.

Contribution

It presents novel asymmetric estimates on geometric and algebraic configurations, enhancing existing bounds in additive combinatorics and decomposition results.

Findings

Improved lower bounds for max{|A-A|, |AA|} and max{|A-A|, |A/A|}

Existence of sets B, C with controlled additive and multiplicative energies

Enhanced bounds in difference-product/division estimates and Balog-Wooley decomposition

Abstract

We show two asymmetric estimates, one on the number of collinear triples and the other on that of solutions to $(a_1+a_2)(a_1^{\prime\prime\prime}+a_2^{\prime\prime\prime})=(a_1^\prime+a_2^\prime)(a_1^{\prime\prime}+a_2^{\prime\prime})$. As applications, we improve results on difference-product/division estimates and on Balog-Wooley decomposition: For any finite subset $A$ of $\mathbb{R}$, \[ \max\{|A-A|,|AA|\} \gtrsim |A|^{1+105/347},\quad \max\{|A-A|,|A/A|\} \gtrsim |A|^{1+15/49}. \] Moreover, there are sets $B,C$ with $A=B\sqcup C$ such that \[ \max\{E^+(B),\, E^\times (C)\} \lesssim |A|^{3-3/11}. \]