On iterated product sets with shifts, II

for any finite $A \subset ℚ$ and any nonzero $u \in ℚ$ . Here, $| A^{(k)} |$ denotes the $k$ -fold product set ${a_{1} \dots a_{k} : a_{1}, \dots, a_{k} \in A}$ .

Furthermore, our method of proof also gives the following $l_{\infty}$ sum-product estimate. For all $γ > 0$ there exists a constant $C = C (γ)$ such that for any $A \subset ℚ$ with $| A A | \leq K | A |$ and any $c_{1}, c_{2} \in ℚ ∖ {0}$ , there are at most $K^{C} | A |^{γ}$ solutions to

c_{1} x + c_{2} y = 1, (x, y) \in A \times 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

Vol. 14, No. 8, 2020

Brandon Hanson, Oliver Roche-Newton and Dmitrii Zhelezov

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors