On parametric geometry of numbers
Volume 195 / 2020
Abstract
Our subject is the study of Minkowski’s Geometry of Numbers for a given lattice and a convex body in depending on a parameter q \ge 0. It arose from questions on Diophantine approximation. The investigations were started by Schmidt and Summerer [Acta Math. 140 (2009), Monatsh. Math. 169 (2013)] and were greatly advanced by work of Roy [Ann. of Math. 182 (2015), Compos. Math. 153 (2017)].
The logarithms of the n successive minima depending on q give rise to a map \boldsymbol L : \mathbb R_{\ge 0} \to \mathbb R^n. In the first two sections, basic properties of \boldsymbol L are stated, and a crucial conjecture is formulated. In later sections more information on \boldsymbol L is gathered, e.g. whether its combined graph is “connected”, on related maps \boldsymbol P with simple properties, and on \liminf and \limsup of the components of the map \boldsymbol L(q)/q.