On uniform boundedness of a rational distance set in the plane
type: Item
instance of: scholarly article
title: On uniform boundedness of a rational distance set in the plane
zbMATH Open document ID: 1239.14016
author: Mehdi Makhul
author: Jafar Shaffaf
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2012-03-22T00:00:00Z
review text: A rational distance set in the plane is a set \(S\) such that the distance of any two points of \(S\) is rational. It is known [\textit{J. Solymosi} and \textit{F. de Zeeuw}, Discrete Comput. Geom. 43, No. 2, 393--401 (2010; Zbl 1209.52009)] that any rational distance set can have only finitely many points in common with an irreducible algebraic curve defined over \({\mathbb R}\), unless the curve is a line or a circle. The so-called weak Lang conjecture states that, if \(X\) is a variety of general type defined over a number field \(K\), then \(X(K)\) is not Zariski dense in \(X\). If true, the weak Lang conjecture implies a uniform bound \(B(K,g)\) for the number of points of a curve of genus \(g\geq 2\) defined over a number field \(K\). Under the weak Lang conjecture, the authors prove that if \(S\) is a rational distance set having finite intersection with each line, then there exists an integer \(B\) such that \(\#(S\cap l)\leq B(\mathbb Q,2)\) for each line \(l\).
reviewed by: Roberto Dvornicich
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 52C10
zbMATH DE Number: 6017082
zbMATH Keywords: rational distance sets
zbMATH Keywords: weak Lang conjecture
MaRDI profile type: Publication
full work available at URL: j.crma.2012.01.010
OpenAlex ID: W2061545969
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DOI: 10.1016/J.CRMA.2012.01.010
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description: scientific article; zbMATH DE number 6017082