Problem 52

Erdős Problem #52: declared status 'open'. Formalized: yes. Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$$$\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?$$ Current best: Erd\H{o}s and Szemer\'{e}di [ErSz83] proved a lower bound of $\lvert A\rvert^{1+c}$ for some constant $c>0$, and an upper bound of$$\lvert A\rvert^2 \exp\left(-c\frac{\log\lvert A\rvert}{\log\log \lvert A\rvert}\right)$$for some constant $c>0$. The lower bound has been improved a number of times. The current record is$$\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{1270}{951}-o(1)}$$due to Bloom [Bl25] (note $1270/951=1.33543\cdots$). The best bound for complex numbers is$$\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{4}{3}+c}$$for some absolute constant $c>0$, due to Basit and Lund [BaLu19]. Prize: $250. OEIS: A263996. Tags: additive combinatorics, number theory.