Vol. 14, No. 8, 2020

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On iterated product sets with shifts, II

Brandon Hanson, Oliver Roche-Newton and Dmitrii Zhelezov

Vol. 14 (2020), No. 8, 2239–2260
DOI: 10.2140/ant.2020.14.2239
Abstract

The main result of this paper is the following: for all b there exists k = k(b) such that

max{|A(k)|,|(A + u)(k)|}|A|b,

for any finite A and any nonzero u . Here, |A(k)| denotes the k-fold product set {a1ak : 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 with |AA| K|A| and any c1,c2 {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 utilize a query-complexity analogue of the polynomial Freiman–Ruzsa conjecture, due to Pälvölgyi and Zhelezov (2020). This new tool replaces the role of the complicated setup of Bourgain and Chang (2004), which we had previously used. Furthermore, there is a better quantitative dependence between the parameters.

Keywords
sum-product problem, S-units, weak Erdős–Szemerédi, unbounded growth conjecture, subspace theorem
Mathematical Subject Classification 2010
Primary: 11B99
Secondary: 11D72
Milestones
Received: 28 January 2020
Revised: 30 March 2020
Accepted: 1 May 2020
Published: 18 September 2020
Authors
Brandon Hanson
Department of Mathematics
University of Georgia
Boyd Graduate Studies Research Center
Athens, GA
United States
Oliver Roche-Newton
Johann Radon Institute for Computational and Applied Mathematics
Linz
Austria
Dmitrii Zhelezov
Alfréd Rényi Institute of Mathematics
Hungarian Academy of Sciences
Budapest
Hungary