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.
"Generalized simultaneous approximation to $m$ linearly dependent reals." Mosc. J. Comb. Number Theory 8 (3) 219 - 228, 2019. https://doi.org/10.2140/moscow.2019.8.219