Problem Statement
Let $A\subset (1,\infty)$ be a countably infinite set such that for all $x\neq y\in A$ and integers $k\geq 1$ we have\[ \lvert kx -y\rvert \geq 1.\]Does this imply that $A$ is sparse? In particular, does this imply that\[\sum_{x\in A}\frac{1}{x\log x}<\infty\]or\[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n)?\]
Categories:
Primitive Sets
Progress
Note that if $A$ is a set of integers then the condition implies that $A$ is a primitive set (that is, no element of $A$ is divisible by any other), for which the convergence of $\sum_{n\in A}\frac{1}{n\log n}$ was proved by Erdős [Er35], and the upper bound\[\sum_{n<x}\frac{1}{n}\ll \frac{\log x}{\sqrt{\log\log x}}\]was proved by Behrend [Be35]. This $O(\cdot)$ bound was improved to a $o(\cdot)$ bound by Erdős, Sárkőzy, and Szemerédi [ESS67].In [Er73] and [Er77c] Erdős mentions an unpublished proof of Haight that\[\lim \frac{\lvert A\cap [1,x]\rvert}{x}=0\]holds if the elements of $A$ are independent over $\mathbb{Q}$.
Over the years Erdős asked for various different quantitative estimates, for example\[\liminf \frac{\lvert A\cap [1,x]\rvert}{x}=0\]or even (motivated by Behrend's bound)\[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}\ll \frac{\log x}{\sqrt{\log\log x}}.\]In [Er97c] he offers \$500 for resolving the questions in the main problem statement above.
This was partially resolved by Koukoulopoulos, Lamzouri, and Lichtman [KLL25], who proved that we must have\[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n).\]See also [858].
This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.
Source: erdosproblems.com/143 | Last verified: January 13, 2026