Dense sets of natural numbers with unusually large least common multiples

TL;DR

This paper constructs special sets of natural numbers with large least common multiples, demonstrating optimal growth properties and answering a question posed by Erdős and Graham, while also clarifying the structure of certain coprime sets.

Contribution

It introduces a novel construction of sets with prescribed LCM and sum properties, resolving an open problem and providing insights into the nature of 'mostly coprime' sets.

Findings

Constructed sets with specified growth of harmonic sums

Established bounds on sums involving least common multiples

Answered a longstanding question of Erdős and Graham

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.