On the size of the set $AA+A$

Oliver Roche-Newton, Imre Z. Ruzsa, Chun-Yen Shen, Ilya D. Shkredov

Published 2018-01-31Version 1

It is established that there exists an absolute constant $c>0$ such that for any finite set $A$ of positive real numbers $$|AA+A| \gg |A|^{\frac{3}{2}+c}.$$ On the other hand, we give an explicit construction of a finite set $A \subset \mathbb R$ such that $|AA+A|=o(|A|^2)$, disproving a conjecture of Balog.