On product sets of arithmetic progressions

We prove that the size of the product set of any finite arithmetic progression $\mathcal{A}\subset \mathbb{Z}$ satisfies $$|\mathcal A \cdot \mathcal A| \ge \frac{|\mathcal A|^2}{(\log |\mathcal A|)^{2\theta +o(1)} } ,$$ where $2\theta=1-(1+\log\log 2)/(\log 2)$ is the constant appearing in the celebrated Erd\H{o}s multiplication table problem. This confirms a conjecture of Elekes and Ruzsa from about two decades ago. If instead $\mathcal{A}$ is relaxed to be a subset of a finite arithmetic progression in integers with positive constant density, we prove that $$|\mathcal A \cdot \mathcal A | \ge \frac{|\mathcal A|^{2}}{(\log |\mathcal A|)^{2\log 2- 1 + o(1)}}.$$ This solves the typical case of another conjecture of Elekes and Ruzsa on the size of the product set of a set $\mathcal{A}$ whose sumset is of size $O(|\mathcal{A}|)$. Our bounds are sharp up to the $o(1)$ term in the exponents. We further prove asymmetric extensions of the above results.