  Given sets $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^2$ of sizes $m$ and
$n$ respectively, we are interested in the number of distinct distances spanned
by $\mathcal{P} \times \mathcal{Q}$. Let $D(m, n)$ denote the minimum number of
distances determined by sets in $\mathbb{R}^2$ of sizes $m$ and $n$
respectively, where $m \leq n$. Elekes \cite{CircleGrids} showed that $D(m, n)
= O(\sqrt{mn})$ when $m \leq n^{1/3}$. For $m \geq n^{1/3}$, we have the upper
bound $D(m, n) = O(n/\sqrt{\log n})$ as in the classical distinct distances
problem.
  In this work, we show that Elekes' construction is tight by deriving the
lower bound of $D(m, n) = \Omega(\sqrt{mn})$ when $m \leq n^{1/3}$. This is
done by adapting Sz\'{e}kely's crossing number argument. We also extend the
Guth and Katz analysis for the classical distinct distances problem to show a
lower bound of $D(m, n) = \Omega(\sqrt{mn}/\log n)$ when $m \geq n^{1/3}$.
