Asymmetric estimates and the sum-product problems
Volume 198 / 2021
Acta Arithmetica 198 (2021), 289-311
MSC: Primary 11B30; Secondary 11B13, 11B75, 05B10.
DOI: 10.4064/aa200803-10-9
Published online: 4 January 2021
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 \[ \max \{E^+(B),\, E^\times (C)\} \lesssim |A|^{3-3/11}. \]
