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Let $A\subset \mathbb{R}$ be a countably infinite set (with $1\not\in A$) 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 \[\sum_{x\in A}\frac{1}{x\log x}<\infty,\] or \[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n)?\]
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 that $\sum_{n<x}\frac{1}{n}=o(\log x)$ was proved by Behrend [Be35].
Additional thanks to: Zachary Chase