Best versus uniform Diophantine approximatio
Martin Rivard-Cooke, Damien Roy
TL;DR
This work studies the spectra of exponents measuring how well $m$-dimensional subspaces of $K_w^n$ can be approximated by $K$-defined subspaces, unifying the Archimedean setting with general number fields via parametric geometry of numbers. By encoding approximation data through $n$-systems and their backwards variants, the authors obtain a duality framework that translates spectral questions into combinatorial questions about piecewise-linear, sign-free $n$-systems. They derive complete or near-complete descriptions for key cases, notably $(\widehat{\omega}_0,\omega_0)$ and $(\widehat{\omega}_{n-2},\omega_{n-2})$, and provide a constructive description for general $(m,n)$ via $n$-systems, including the density of self-similar instances and the duality with backwards systems. A striking finding is the failure of a natural conjecture for $S_{3,5}$, where the boundary appears non-semialgebraic, illustrating the richness and complexity of spectra in higher dimensions. Overall, the paper advances the understanding of spectra in Diophantine approximation by delivering new proofs, duality principles, and explicit boundary phenomena using PGN techniques, with implications for number-field settings and potential generalizations to broader classes of $n$-systems.
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 $ω_{m-1}(\textbf{u})$ denote the exponent of best approximation to $\textbf{u}$ by vector subspaces of $K_w^n$ of dimension $m$ defined over $K$, and let $\widehatω_{m-1}(\textbf{u})$ denote the corresponding exponent of uniform approximation. Finally, let $S_{m,n}$ denote the set of all pairs $(\widehatω_{m-1}(\textbf{u}),ω_{m-1}(\textbf{u}))$ 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 $(m,n)$.