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Abstract
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In order to analyse the simultaneous approximation properties of
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
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
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Mathematical Subject Classification 2010
Primary: 11H06, 11J13
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Milestones
Received: 17 October 2018
Revised: 20 March 2019
Accepted: 22 May 2019
Published: 23 July 2019
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