2018/06/05 by Hanson, Brandon, Roche-Newton, Oliver, Zhelezov, Dmitrii · 1 citation
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1806.01697
The main result of this paper is the following: for all b ∈ \mathbb Z there exists k=k(b) such that max \ |A(k)|, |(A+u)(k)| \ ≥ |A|b, for any finite A ⊂ \mathbb Q and any non-zero u ∈ \mathbb Q. Here, |A(k)| denotes the k-fold product set \a1⋯ ak : a1, …, ak ∈ A \. Furthermore, our method of proof also gives the following l∞ sum-product estimate. For all γ>0 there exists a constant C=C(γ) such that for any A ⊂ \mathbb Q with |AA| ≤ K|A| and any c1,c2 ∈ \mathbb Q ∖ \0\, there are at most KC|A|γ solutions to c1x + c2y =1 , (x,y) ∈ A × A. In particular, this result gives a strong bound when K=|A|ε, provided that ε>0 is sufficiently small, and thus improves on previous bounds obtained via the Subspace Theorem. In further applications we give a partial structure theorem for point sets which determine many incidences and prove that sum sets grow arbitrarily large by taking sufficiently many products. We utilise a query-complexity analogue of the polynomial Freiman-Ruzsa conjecture, due to Zhelezov and Pálvölgyi. This new tool replaces the role of the complicated setup of Bourgain and Chang, which we had previously used. Furthermore, there is a better quantitative dependence between the parameters.