OPEN

Let $h(n)$ be such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set
\[\left\{ \frac{a}{(a,b)}: a,b\in A\right\}\]
has size at least $h(n)$. Estimate $h(n)$.

Erdős and Szemerédi proved that
\[n^{1/2} \ll h(n) \ll n^{1-c}\]
for some constant $c>0$.