OPEN - $500
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
\[\liminf \frac{\lvert A\cap [1,x]\rvert}{x}=0?\]
Or
\[\sum_{x\in A}\frac{1}{x\log x}<\infty,\]
or
\[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}=o(\log n)?\]
Perhaps even
\[\sum_{\substack{x <n\\ x\in A}}\frac{1}{x}\ll \frac{\log x}{\sqrt{\log\log x}}?\]
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In [Er73] mentions an unpublished proof of Haight that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 $\mathbb{Q}$.
See also [858].
