#### 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 $b\in ℤ$ 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.

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