Few distinct distances implies no heavy lines or circles

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.

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Correspondence to Adam Sheffer.

Additional information

The first author was partially supported by Grant 338/09 from the Israel Science Fund and by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11). Work by the second author was partially supported by the Department of Defense through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. Work by the third author was partially supported by Swiss National Science Foundation Grant no. 200021-513047.

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Sheffer, A., Zahl, J. & de Zeeuw, F. Few distinct distances implies no heavy lines or circles. Combinatorica 36, 349–364 (2016). https://doi.org/10.1007/s00493-014-3180-6

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Mathematics Subject Classication (2000)

  • 52C10