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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 that $\sum_{n<x}\frac{1}{n}=o(\log x)$ was proved by Behrend [Be35].

In [Er73] mentions an unpublished proof of Haight that \[\lim \frac{\lvert A\cap [1,x]\rvert}{x}=0\] holds if the elements of $A$ are independent over $\mathbb{Q}$.

Additional thanks to: Zachary Chase
A set $A\subset \mathbb{N}$ is primitive if no member of $A$ divides another. Is the sum \[\sum_{n\in A}\frac{1}{n\log n}\] maximised over all primitive sets when $A$ is the set of primes?
Erdős [Er35] proved that this sum always converges for a primitive set. Solved by Lichtman [Li23].
Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let \[B = \{ m\in \mathbb{N} : m\not\in X_n\pmod{n}\textrm{ for all }n\in A\}.\] Must $B$ have a logarithmic density, i.e. is it true that \[\lim_{x\to \infty} \frac{1}{\log x}\sum_{\substack{m\in B\\ m<x}}\frac{1}{m}\] exists?
Davenport and Erdős [DaEr37] proved that the answer is yes when $X_n=\{0\}$ for all $n\in A$. The problem considers logarithmic density since Besicovitch [Be34] showed examples exist without a natural density, even when $X_n=\{0\}$ for all $n\in A$.