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on Erdős #959 · geometry / distances

The statement

Let $A\subset \mathbb{R}^2$ 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, and suppose the $d_i$ are ordered such 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 $n$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.