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Discrete Analysis Seminar

Tuesday, November 15, 2022
3:00pm to 4:00pm
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Linde Hall 255
Product sets of arithmetic progressions
Wenqiang Xu, Department of Mathematics, Stanford University,

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 sum set is of size $O(|\mathcal{A}|)$.

Our bounds are sharp up to the $o(1)$ term in the exponents.

This is joint work with Yunkun Zhou.

For more information, please contact Math Department by phone at 626-395-4335 or by email at [email protected].