A problem of Erd\H{o}s about rich distances

TL;DR

This paper addresses Erdős's old question by constructing point sets with many distances occurring more than a linear number of times, providing affirmative answers and generalizations to the problem.

Contribution

The paper constructs explicit point sets demonstrating that many distances can occur more than n times, answering Erdős's question affirmatively and generalizing the result.

Findings

Existence of point sets with loor(n/4)istances occurring more than n times

Generalization to sets with c_m  n distances occurring more than n+m times

Provides explicit constructions for these point sets

Abstract

An old question posed by Erd\H{o}s asked whether there exists a set of $n$ points such that $c \cdot n$ distances occur more than $n$ times. We provide an affirmative answer to this question, showing that there exists a set of $n$ points such that $\lfloor \frac{n}{4}\rfloor$ distances occur more than $n$ times. We also present a generalized version, finding a set of $n$ points where $c_m \cdot n$ distances occurring more than $n+m$ times.