  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.
