Problem 143

Erdős Problem #143: declared status 'open'. Formalized: yes. 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$$\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n)?$$ Current best: 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 Erd\H{o}s [Er35], and the upper bound$$\sum_{n<x}\frac{1}{n}\ll \frac{\log x}{\sqrt{\log\log x}}$$was proved by Behrend [Be35]. Prize: $500. Tags: primitive sets.