Best versus uniform Diophantine approximatio

Abstract

Let $0<m<n$ be integers, and let $K_w$ denote the completion of a number field $K$ at a non-trivial place $w$. For each non-zero $\textbf{u}\in K_w^n$, let $\omega_{m-1}(\textbf{u})$ denote the exponent of best approximation to $\textbf{u}$ by vector subspaces of $K_w^n$ of dimension $m$ defined over $K$, and let $\widehat{\omega}_{m-1}(\textbf{u})$ denote the corresponding exponent of uniform approximation. Finally, let $S_{m,n}$ denote the set of all pairs $(\widehat{\omega}_{m-1}(\textbf{u}),\omega_{m-1}(\textbf{u}))$ where $\textbf{u}$ runs through all points of $K_w^n$ with linearly independent coordinates over $K$. In this paper we use parametric geometry of numbers to study this spectrum $S_{m,n}$, noting at first that it is independent of the choice of $K$ and $w$. We may thus assume that $K=\mathbb{Q}$ and $K_w=\mathbb{R}$. In this context, Schmidt and Summerer proposed conjectural descriptions for $S_{1,n}$ and $S_{n-1,n}$ which were confirmed by Marnat and Moshchevitin for each $n\ge 2$. We give an alternative proof of their result based on the PhD thesis of the first author, highlighting the duality between the two spectra. In his thesis, the first author generalized the conjecture to any pair $(m,n)$ and proved it to be true also for $S_{2,4}$. We present this as well, but show that this natural conjecture fails for $S_{3,5}$. Moreover, the part of $S_{3,5}$ that we succeed to compute here suggests a complicated boundary for that set, possibly not semialgebraic. We also give a qualitative description of $S_{m,n}$ for a general pair $(m,n)$.