Erdős problem 959

Let $A\subseteq \mathbb{R}^2$ be a set of size $n$ and let $\{d_1,\ldots,d_k\}$ be the set of distinct distances determined by $A$. Let $f(d)$ be the number of times the distance $d$ is determined, ordered so that $f(d_1)\geq f(d_2)\geq \cdots \geq f(d_k)$. Estimate $$\max (f(d_1)-f(d_2)),$$ where the maximum is taken over all $A$ of size $n$ (this is extremalGap n).

The asymptotic order of extremalGap is not known; a natural formalization of "estimate" asks whether it has a well-defined polynomial growth exponent.