## Abstract

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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## Notes

- 1.
The term

*additive energy*, referring to the number of quadruples (*a*,*b*,*c*,*d*) in some underlying set of numbers such that \(a+b = c+d\), was coined by Tao and Vu [8]. Starting with the work of Elekes and Sharir [3], and Guth and Katz [6] on the distinct distance problem, the strategy of using geometric incidence bounds to obtain upper bounds on analogously defined energies has become indispensable in the study of questions about the number of distinct equivalent subsets.

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## Acknowledgements

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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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). https://doi.org/10.1007/s00026-019-00478-z

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### Keywords

- Incidences
- Perpendicular bisectors
- Distinct distances
- Energy bound

### Mathematics Subject Classification

- 52C10
- 05D99