TY - JOUR

T1 - On distinct perpendicular bisectors and pinned distances in finite fields

AU - Hanson, Brandon

AU - Lund, Ben

AU - Roche-Newton, Oliver

N1 - Funding Information:
Brandon Hanson was supported by NSERC of Canada. Ben Lund was supported by NSF grant CCF-1350572 . Oliver Roche-Newton was supported by the Austrian Science Fund (FWF): Project F5511-N26 , which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. Part of this research was undertaken when the authors were visiting the Institute for Pure and Applied Mathematics, UCLA, which is funded by NSF grant DMS-0931852 . We are grateful to Swastik Kopparty, Doowon Koh, Tom Robbins, Adam Sheffer and Frank de Zeeuw for several helpful conversations related to the content of this paper. Finally, we are grateful to an anonymous referee for several comments which have helped to improve the exposition of the paper.
Publisher Copyright:
© 2015 Elsevier Inc. All rights reserved.

PY - 2016/1/1

Y1 - 2016/1/1

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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U2 - 10.1016/j.ffa.2015.10.002

DO - 10.1016/j.ffa.2015.10.002

M3 - Article

AN - SCOPUS:84946102745

VL - 37

SP - 240

EP - 264

JO - Finite Fields and Their Applications

JF - Finite Fields and Their Applications

SN - 1071-5797

ER -