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2020 On iterated product sets with shifts, II
Brandon Hanson, Oliver Roche-Newton, Dmitrii Zhelezov
Algebra Number Theory 14(8): 2239-2260 (2020). DOI: 10.2140/ant.2020.14.2239

Abstract

The main result of this paper is the following: for all $b\in ℤ$ there exists $k=k\left(b\right)$ such that

$max\left\{|{A}^{\left(k\right)}|,|{\left(A+u\right)}^{\left(k\right)}|\right\}\ge |A{|}^{b},$

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

Furthermore, our method of proof also gives the following ${l}_{\infty }$ sum-product estimate. For all $\gamma >0$ there exists a constant $C=C\left(\gamma \right)$ such that for any $A\subset ℚ$ with $|AA|\le K|A|$ and any ${c}_{1},{c}_{2}\in ℚ\setminus \left\{0\right\}$, there are at most ${K}^{C}|A{|}^{\gamma }$ solutions to

${c}_{1}x+{c}_{2}y=1,\phantom{\rule{1em}{0ex}}\left(x,y\right)\in A×A.$

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.

Citation

Brandon Hanson. Oliver Roche-Newton. Dmitrii Zhelezov. "On iterated product sets with shifts, II." Algebra Number Theory 14 (8) 2239 - 2260, 2020. https://doi.org/10.2140/ant.2020.14.2239

Information

Received: 28 January 2020; Revised: 30 March 2020; Accepted: 1 May 2020; Published: 2020
First available in Project Euclid: 12 November 2020

MathSciNet: MR4172707
Digital Object Identifier: 10.2140/ant.2020.14.2239

Subjects:
Primary: 11B99
Secondary: 11D72

Keywords: subspace theorem , sum-product problem , S-units , unbounded growth conjecture , weak Erdős–Szemerédi