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Note on the Number of Hinges Defined by a Point Set in ℝ2

Abstract

It is shown that the number of distinct types of three-point hinges, defined by a real plane set of n points is ≫n2 log−3n, where a hinge is identified by fixing two pairwise distances in a point triple. This is achieved via strengthening (modulo a logn factor) of the Guth- Katz estimate for the number of pairwise intersections of lines in ℝ3, arising in the context of the plane Erdős distinct distance problem, to a second moment incidence estimate. This relies, in particular, on the generalisation of the Guth-Katz incidence bound by Solomon and Sharir.

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Correspondence to Misha Rudnev.

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Partially supported by the Leverhulme Trust Grant RPG-2017-371.

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Rudnev, M. Note on the Number of Hinges Defined by a Point Set in ℝ2. Combinatorica 40, 749–757 (2020). https://doi.org/10.1007/s00493-020-4171-4

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