  In this note it is established that, for any finite set $A$ of real numbers,
there exist two elements $a,b \in A$ such that
  $$|(a+A)(b+A)| \gg \frac{|A|^2}{\log |A|}.$$
  In particular, it follows that $|(A+A)(A+A)| \gg \frac{|A|^2}{\log |A|}$. The
latter inequality had in fact already been established in an earlier work of
the author and Rudnev (arXiv:1203.6237), which built upon the recent
developments of Guth and Katz (arXiv:1011.4105) in their work on the Erd\H{o}s
distinct distance problem. Here, we do not use those relatively deep methods,
and instead we need just a single application of the Szemer\'{e}di-Trotter
Theorem. The result is also qualitatively stronger than the corresponding
sum-product estimate from (arXiv:1203.6237), since the set $(a+A)(b+A)$ is
defined by only two variables, rather than four. One can view this as a
solution for the pinned distance problem, under an alternative notion of
distance, in the special case when the point set is a direct product $A \times
A$. Another advantage of this more elementary approach is that these results
can now be extended for the first time to the case when $A \subset \mathbb C$.
