A Refined Energy Bound for Distinct Perpendicular Bisectors


Let \({\mathcal {P}}\) be a set of n points in the Euclidean plane. We prove that, for any \(\varepsilon > 0\), either a single line or circle contains n/2 points of \({\mathcal {P}}\), or the number of distinct perpendicular bisectors determined by pairs of points in \({\mathcal {P}}\) is \(\Omega (n^{52/35 - \varepsilon })\), where the constant implied by the \(\Omega \) notation depends on \(\varepsilon \). This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of \({\mathcal {P}}\), or the number of distinct perpendicular bisectors is \(\Omega (n^2)\). The proof relies bounding the size of a carefully selected subset of the quadruples \((a,b,c,d) \in {\mathcal {P}}^4\) such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d.

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I thank Brandon Hanson, Peter Hajnal, Oliver Roche-Newton, Adam Sheffer, and Frank de Zeeuw for many stimulating conversations on perpendicular bisectors and related questions. I thank Luca Ghidelli for pointing out an error in Lemma 7 in an earlier version. I thank the anonymous referees for numerous helpful comments on the writing and presentation of this paper.

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Correspondence to Ben Lund.

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This work was supported by ERC Grant 267165 DISCONV.

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Lund, B. A Refined Energy Bound for Distinct Perpendicular Bisectors. Ann. Comb. (2020).

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  • Incidences
  • Perpendicular bisectors
  • Distinct distances
  • Energy bound

Mathematics Subject Classification

  • 52C10
  • 05D99