Dense sets of natural numbers with unusually large least common multiples

Abstract

For any constant $C_0>0$, we construct a set $A \subset {\mathbb N}$ such that one has $$ \sum_{n \in A: n \leq x} \frac{1}{n} = \exp\left(\left(\frac{C_0}{2}+o(1)\right) (\log\log x)^{1/2} \log\log\log x \right)$$ and $$ \sum_{n,m \in A: n, m \leq x} \frac{1}{\operatorname{lcm}(n,m)} \ll_{C_0} \left(\sum_{n \in A: n \leq x} \frac{1}{n}\right)^2$$ as $x \to \infty$, with the growth rate given here optimal up to the dependence on $C_0$. This answers in the negative a question of Erd\H{o}s and Graham, and also clarifies the nature of certain ``mostly coprime'' sets studied by Bergelson and Richter.