2019/05/09 by Roche-Newton, Oliver, Warren, Audie
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1905.03701
The purpose of this article is to further explore how the structure of the affine group can be used to deduce new incidence theorems, and to explore sum-product type applications of these incidence bounds, building on the recent work of Rudnev and Shkredov. We bound the energy of several systems of lines, in some cases obtaining a better energy bound than the corresponding bounds obtained by Rudnev and Shkredov by exploiting a connection with collinear quadruples. Our motivation for seeking to generalise and improve the incidence bound obtained by Rudnev and Shkredov comes from possible applications to sum-product problems. For example, we prove that, for any finite A ⊂ \mathbb R the following superquadratic bound holds: | \ (ab-cd)/(a-c) : a,b,c,d ∈ A \ | ≫ |A|2+(1)/(14). This improves the previously known bound with exponent 2. We also give a threshold-beating asymmetric sum-product estimate for sets with small sum set by proving that there exists a positive constant c such that for all finite A,B ⊂ \mathbb R, |A+A| ≪ K|A| ⇒ |AB| ≫K |A||B|1/2+c.