Erdős Problem #143 — 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?

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 $\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n)?$