On iterated product sets with shifts II

Abstract

The main result of this paper is the following: for all $b\in \mathbb{Z}$ there exists $k=k(b)$ such that \[ \max \{ |A^{(k)}|, |(A+u)^{(k)}| \} \geq |A|^b, \] for any finite $A\subset \mathbb{Q}$ and any non-zero $u\in \mathbb{Q}$. Here, $|A^{(k)}|$ denotes the $k$-fold product set $\{a_1\cdots a_k:a_1,\dots ,a_k\in A\}$. Furthermore, our method of proof also gives the following $l_{\infty }$ sum-product estimate. For all $\gamma >0$ there exists a constant $C=C(\gamma )$ such that for any $A\subset \mathbb{Q}$ with $|AA|\le K|A|$ and any $c_1,c_2\in \mathbb{Q}\setminus \{0\}$, there are at most $K^C|A{|}^{\gamma }$ solutions to \[ c_1x + c_2y =1 ,\,\,\,\,\,\,\, (x,y) \in A \times A. \] In particular, this result gives a strong bound when $K=|A{|}^{\epsilon }$, provided that $\epsilon >0$ 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.

This paper has shortened considerably, as a consequence of an application of a new result of Zhelezov and Pálvölgyi, see arXiv:2003.04648. This version will appear in Algebra and Number Theory. This paper is a sequel to arXiv:1801.07982, although it can be read independently and does not depend on results from the original paper