Dense sets of natural numbers with unusually large least common multiples
Terence Tao
TL;DR
This work settles Erdős–Graham Problem #442 by showing that one can construct dense sets of natural numbers whose reciprocal-sum growth is as large as $\exp\left( (C_0/2+o(1)) ( {\operatorname{Log}}_2 x)^{1/2} {\operatorname{Log}}_3 x \right)$ while keeping the average gcd-smallness (and hence lcm’s) under a controlled bound $\sum_{n,m\le x} 1/\mathrm{lcm}(n,m) \ll_{C_0} (e^{C_0^2}-1+o(1)) \left( \sum_{n\le x} 1/n \right)^2$. The construction, based on probabilistic models for prime divisors and a refined product-structure of integers, yields near-optimal growth rates (up to the constant depending on $C_0$) and demonstrates a negative answer to the Erdős–Graham question. The analysis also clarifies the nature of Bergelson–Richter’s mostly coprime sets and connects to Gauss’s gcd identity, log-density results for prime factors, and the anatomy of integers. Overall, the paper advances the understanding of how dense sets can be while maintaining unusually large least common multiples on average, by combining probabilistic methods with finely tuned combinatorial constructions across logarithmic scales.
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 Erdős and Graham, and also clarifies the nature of certain ``mostly coprime'' sets studied by Bergelson and Richter.