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Given sets P, Q ⊆ R 2 of sizes m and n respectively, we are interested in the number of distinct distances spanned by P × Q. Let D(m, n) denote the minimum number of distances determined by sets in R 2 of sizes m and n respectively, where m ≤ n.In this work, we show that Elekes' construction is tight by deriving the lower bound of D(m, n) = Ω( √ mn) when m ≤ n 1/3 . This is done by adapting Székely'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) = Ω( √ mn/ log n) when m ≥ n 1/3 .

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 showed that $D(m, n) = O(\sqrt{mn})$ when $m \leqslant n^{1/3}$. For $m \geqslant n^{1/3}$, 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 $m \leqslant n^{1/3}$. This is done by adapting Székely'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 $m \geqslant n^{1/3}$.

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