Erdős problem 650

Let $f(m)$ be such that if $A\subseteq \{1,\ldots,N\}$ has $\lvert A\rvert=m$ then every interval in $[1,\infty)$ of length $2N$ contains $\geq f(m)$ many distinct integers $b_1,\ldots,b_r$ where each $b_i$ is divisible by some $a_i\in A$, where $a_1,\ldots,a_r$ are distinct.

Estimate $f(m)$.

GPT 5.4 Pro (prompted by He, Li, and Tang) proved $f(m)\leq \lceil 2\sqrt{m}\rceil$. A corresponding lower bound was given by GPT 5.4 Pro and Aristotle; it is now known (see the paper of van Doorn, Li, and Tang [VLT26]) that $$f(m) = \min(m, \lceil 2\sqrt{m}\rceil)$$ for all $m$.