### Abstract

Given a set of points P ⊂ F_{q}^{2} such that |P| ≥ q^{4/3}, we establish that for a positive proportion of points a ∈ P, we have|{∥a-b∥:b∈P}|蠑q, where ∥a-b∥ is the distance between points a and b. This improves a result of Chapman et al. [6]. A key ingredient of our proof also shows that, if |P| ≥ q^{3/2}, then the number B of distinct lines which arise as the perpendicular bisector of two points in P satisfies B 蠑 q^{2}.

Original language | English (US) |
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Pages (from-to) | 240-264 |

Number of pages | 25 |

Journal | Finite Fields and their Applications |

Volume | 37 |

DOIs | |

State | Published - Jan 1 2016 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Algebra and Number Theory
- Engineering(all)
- Applied Mathematics

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## Cite this

Hanson, B., Lund, B., & Roche-Newton, O. (2016). On distinct perpendicular bisectors and pinned distances in finite fields.

*Finite Fields and their Applications*,*37*, 240-264. https://doi.org/10.1016/j.ffa.2015.10.002