# Few distinct distances implies no heavy lines or circles

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## Abstract

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set \(\mathcal{P}\) of *n* points determines *o*(*n*) distinct distances, then no line contains *Ω*(*n* ^{7/8}) points of \(\mathcal{P}\) and no circle contains *Ω*(*n* ^{5/6}) points of \(\mathcal{P}\).

We rely on the partial variant of the Elekes-Sharir framework that was introduced by Sharir, Sheffer, and Solymosi in [19] for bipartite distinct distance problems. To prove our bound for the case of lines we combine this framework with a theorem from additive combinatorics, and for our bound for the case of circles we combine it with some basic algebraic geometry and a recent incidence bound for plane algebraic curves by Wang, Yang, and Zhang [20].

A significant difference between our approach and that of [19] (and of other related results) is that instead of dealing with distances between two point sets that are restricted to one-dimensional curves, we consider distances between one set that is restricted to a curve and one set with no restrictions on it.

## Mathematics Subject Classication (2000)

52C10## Preview

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