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Note on the Number of Hinges Defined by a Point Set in ℝ2
Combinatorica ( IF 1.1 ) Pub Date : 2020-05-22 , DOI: 10.1007/s00493-020-4171-4
Misha Rudnev

It is shown that the number of distinct types of three-point hinges, defined by a real plane set of n points is ≫ n 2 log −3 n , 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 ℝ 3 , 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.

中文翻译:

关于ℝ2中点集定义的铰链数量的注意事项

结果表明,由 n 个点的实平面集定义的不同类型的三点铰链的数量为 ≫ n 2 log -3 n ,其中铰链通过固定点三元组中的两个成对距离来识别。这是通过将 ℝ 3 中线的成对交点数量的 Guth-Katz 估计(在平面 Erdő 的不同距离问题的背景下)加强(以 logn 因子为模)来实现的,以进行二阶矩关联估计。这尤其依赖于 Solomon 和 Sharir 约束的 Guth-Katz 关联的概括。
更新日期:2020-05-22
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