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 $$\sum _{n\in A:n\le x}\frac{1}{n}=\exp \left(\left(\frac{C_0}{2}+o(1)\right)(\log \log x{)}^{1/2}\log \log \log x\right)$$ and $$\sum _{n,m\in A:n,m\le x}\frac{1}{\mathrm{lcm}(n,m)}{\ll }_{C_0}{\left(\sum _{n\in A:n\le x}\frac{1}{n}\right)}^2$$ 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 Erdős and Graham, and also clarifies the nature of certain ``mostly coprime'' sets studied by Bergelson and Richter.

20 pages, no figures. Adds an appendix providing an argument of Will Sawin that improved the upper bound to match the lower bound (after correcting a typo in the latter)