On iterated product sets with shifts

We prove that, for any finite set $A \subset \mathbb Q$ with $|AA| \leq K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound $$| \{(a_1+1)(a_2+1) \cdots (a_k+1) : a_i \in A \}| \geq \frac{|A|^k}{(8k^4)^{kK}}. $$ This result is essentially optimal when $K$ is of the order $c\log|A|$, for a sufficiently small constant $c=c(k)$. Our main tool is a multiplicative variant of the $\Lambda$-constants used in harmonic analysis, applied to Dirichlet polynomials.