Generalized simultaneous approximation to $m$ linearly dependent reals

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 $\mathbb{Q}$ 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.