Asymmetric estimates and the sum-product problems

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 \[ \max\{E^+(B),\, E^\times (C)\} \lesssim |A|^{3-3/11}. \]