Estimate $f(m)$. In particular is it true that $f(m)\ll m^{1/2}$?

OPEN

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 $f(m)$. In particular is it true that $f(m)\ll m^{1/2}$?

Erdős and Sarányi [ErSa59] proved that $f(m)\gg m^{1/2}$.