Vol. 8, No. 3, 2019

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Generalized simultaneous approximation to $m$ linearly dependent reals

Leonhard Summerer

Vol. 8 (2019), No. 3, 219–228
Abstract

In order to analyse the simultaneous approximation properties of m reals, the parametric geometry of numbers studies the joint behaviour of the successive minima functions with respect to a one-parameter family of convex bodies and a lattice defined in terms of the m given reals. For simultaneous approximation in the sense of Dirichlet, the linear independence over of these reals together with 1 is equivalent to a certain nice intersection property that any two consecutive minima functions enjoy. This paper focusses on a slightly generalized version of simultaneous approximation where this equivalence is no longer in place and investigates conditions for that intersection property in the case of linearly dependent irrationals.

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Keywords
parametric geometry of numbers, successive minima, simultaneous approximation
Mathematical Subject Classification 2010
Primary: 11H06, 11J13
Milestones
Received: 17 October 2018
Revised: 20 March 2019
Accepted: 22 May 2019
Published: 23 July 2019
Authors
Leonhard Summerer
Faculty of Mathematics
University of Vienna
Vienna
Austria