动力系统研讨班：On product sets of arithmetic progressions

中科院数学与系统科学研究院

数学研究所

学术报告

动力系统研讨班

报告人：Max Wenqiang Xu (Stanford University)

题  目：On product sets of arithmetic progressions

时  间：2023.07.21（星期五）13:45-14:45

地  点：数学院南楼N913

摘  要：

We prove that the size of the product set of any finite arithmetic progression $A$ in integers satisfies $|AA| \ge \frac{|A|^2}{(\log |A|)^{2\theta +o(1)} }$, 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 $A$ is relaxed to be a subset of a finite arithmetic progression in integers with positive constant density, we prove that $|A A | \ge \frac{|A|^{2}}{(\log |A|)^{2\log 2- 1 + o(1)}}$. This solves the typical case of another conjecture of Elekes and Ruzsa on the size of the product set of a set $A$ whose sum set is of size  $O(|A|)$.