On iterated product sets with shifts II

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 $|AA| \leq K|A|$ 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 P\'alv\"olgyi. 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.