On distinct perpendicular bisectors and pinned distances in finite fields

Brandon Hanson; Ben Lund; Oliver Roche-Newton

doi:10.1016/j.ffa.2015.10.002

On distinct perpendicular bisectors and pinned distances in finite fields

Brandon Hanson, Ben Lund, Oliver Roche-Newton

Mathematics

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

Given a set of points P ⊂ F_q² 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 q².

Original language	English (US)
Pages (from-to)	240-264
Number of pages	25
Journal	Finite Fields and their Applications
Volume	37
DOIs	https://doi.org/10.1016/j.ffa.2015.10.002
State	Published - Jan 1 2016

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Algebra and Number Theory
General Engineering
Applied Mathematics

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title = "On distinct perpendicular bisectors and pinned distances in finite fields",

abstract = "Given a set of points P ⊂ Fq2 such that |P| ≥ q4/3, 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| ≥ q3/2, then the number B of distinct lines which arise as the perpendicular bisector of two points in P satisfies B 蠑 q2.",

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AU - Lund, Ben

AU - Roche-Newton, Oliver

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N2 - Given a set of points P ⊂ Fq2 such that |P| ≥ q4/3, 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| ≥ q3/2, then the number B of distinct lines which arise as the perpendicular bisector of two points in P satisfies B 蠑 q2.

AB - Given a set of points P ⊂ Fq2 such that |P| ≥ q4/3, 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| ≥ q3/2, then the number B of distinct lines which arise as the perpendicular bisector of two points in P satisfies B 蠑 q2.

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ER -

On distinct perpendicular bisectors and pinned distances in finite fields

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