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.

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