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 $\mathbb R^n$ 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 $\boldsymbol L(q)/q$.
