Erdős Problem #490

Let \(A,B\subseteq \{1,\ldots,N\}\) be such that all the products \(ab\) with \(a\in A\) and \(b\in B\) are distinct. Is it true that \[\lvert A\rvert \lvert B\rvert \ll \frac{N^2}{\log N}?\] This would be best possible, for example letting \(A=[1,N/2]\cap \mathbb{N}\) and \(B=\{ N/2<p\leq N: p\textrm{ prime}\}\). This is true, and was proved by Szemerédi [Sz76]. In [Er72] Erdős goes on to ask whether \[\lim_{N\to \infty}\max_{A,B\subseteq [N]}\frac{\lvert A\rvert\lvert B\rvert\log N}{N^2}\] exists, where the maximum is over \(A\) and \(B\) with all the products \(ab\) distinct, and to determine its value. As noted in the comments to [896](https://www.erdosproblems.com/896) by van Doorn, if the limit exists it must be \(\geq 1\). See also [425](https://www.erdosproblems.com/425) and [896](https://www.erdosproblems.com/896).