On uniform boundedness of a rational distance set in the plane (Q765735)

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 \(l\).