Joint approximations of real and \(p\)-adic numbers by algebraic integers. (Russian. English summary) Zbl 1247.11094

Zorin, Evgeniy V.; Budarina, Natalia V.

Vestn. Beloruss. Gos. Univ., Ser. 1, Fiz. Mat. Inform. 2009, No. 2, 104-109 (2009).

Summary: We consider the problem of joint approximations of two points from the fields \(\mathbb C\) and \({\mathbb Q}_ p\), respectively, by two algebraic integers. The distance between these numbers is determined by the absolute and \(p\)-adic metrics. As an auxiliary result, we prove a version of Minkowski’s theorem on consecutive minima taking into account the inequality in the \(p\)-adic metric along with the inequalities in the absolute metric.

MSC:

11J13	Simultaneous homogeneous approximation, linear forms
11J61	Approximation in non-Archimedean valuations