On distinct perpendicular bisectors and pinned distances in finite fields

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• 作者：Brandon Hanson^a ; ^{bwh5339@psu.edu" class="auth_mail" title="E-mail the corresponding author} ; Ben Lund^b ; ^{lund.ben@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Oliver Roche-Newton^c ; ^{o.rochenewton@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：52C10
• 刊名：Finite Fields and Their Applications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：37
• 期：Complete
• 页码：240-264
• 全文大小：443 K

文摘

Given a set of points $P\subset {\mathbb{F}}_q^2$ such that |P|≥q^4/3$|P|\ge q^{4/3}$, we establish that for a positive proportion of points a07377a7057e4" title="Click to view the MathML source">a∈P$\mathit{a}\in P$, we have where ‖a−b‖$\Vert \mathit{a}-\mathit{b}\Vert$ 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$|P|\ge q^{3/2}$, then the number B of distinct lines which arise as the perpendicular bisector of two points in P   satisfies B≫q²$B\gg q^2$.