Bisector Energy and Few Distinct Distances

Abstract

We define the bisector energy $$\mathcal E(\mathcal P)$$ of a set $$\mathcal P$$ in $$\mathbb R^2$$ to be the number of quadruples $$(a,b,c,d)\in \mathcal P^4$$ such that ab determine the same perpendicular bisector as cd. Equivalently, $$\mathcal E(\mathcal P)$$ is the number of isosceles trapezoids determined by $$\mathcal P$$. We prove that for any $$\varepsilon >0$$, if an n-point set $$\mathcal P$$ has no M(n) points on a line or circle, then we have

\begin{aligned} \mathcal E(\mathcal P) = O\big (M(n)^{\frac{2}{5}}n^{\frac{12}{5}+\varepsilon } + M(n)n^2\big ). \end{aligned}

We derive the lower bound $$\mathcal E(\mathcal P)=\Omega (M(n)n^2)$$, matching our upper bound when M(n) is large. We use our upper bound on $$\mathcal E(\mathcal P)$$ to obtain two rather different results:

1. (i)

If $$\mathcal P$$ determines $$O(n/\sqrt{\log n})$$ distinct distances, then for any $$0<\alpha \le 1/4$$, there exists a line or circle that contains at least $$n^\alpha$$ points of $$\mathcal P$$, or there exist $$\Omega (n^{8/5-12\alpha /5-\varepsilon })$$ distinct lines that contain $$\Omega (\sqrt{\log n})$$ points of $$\mathcal P$$. This result provides new information towards a conjecture of Erdős (Discrete Math 60:147–153, 1986) regarding the structure of point sets with few distinct distances.

2. (ii)

If no line or circle contains M(n) points of $$\mathcal P$$, the number of distinct perpendicular bisectors determined by $$\mathcal P$$ is $$\Omega \left( \min \left\{ M(n)^{-2/5}n^{8/5-\varepsilon }, M(n)^{-1} n^2\right\} \right)$$.

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Notes

1. Throughout this paper, when we state a bound involving an $$\varepsilon$$, we mean that this bound holds for every $$\varepsilon >0$$, with the multiplicative constant of the O()-notation depending on $$\varepsilon$$.

2. We define the dimension of a real algebraic variety as in [3, Sect. 2.8].

3. This lemma only applies to complex varieties. However, we can take the complexification of the real variety and apply the lemma to it (for the definition of a complexification, see for example [27, Sect. 10]). The number of irreducible components of the complexification cannot be smaller than number of irreducible components of the real variety (see for instance [27, Lem. 7]).

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Acknowledgments

Part of this research was performed while the authors were visiting the Institute for Pure and Applied Mathematics (IPAM) in Los Angeles, which is supported by the National Science Foundation. Work on this paper by Frank de Zeeuw was partially supported by Swiss National Science Foundation Grants 200020-144531 and 200021-137574. Work on this paper by Ben Lund was supported by NSF Grant CCF-1350572.

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Lund, B., Sheffer, A. & de Zeeuw, F. Bisector Energy and Few Distinct Distances. Discrete Comput Geom 56, 337–356 (2016). https://doi.org/10.1007/s00454-016-9783-5

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• DOI: https://doi.org/10.1007/s00454-016-9783-5

Keywords

• Discrete geometry
• Incidence geometry
• Polynomial method
• Distinct distances
• Perpendicular bisectors