Bad approximability, bounded ratios and Diophantine exponents

TL;DR

The paper investigates how bounded ratios of successive best Diophantine approximations for real matrices ${\boldsymbol\xi}\in\mathbb{R}^{n\times m}$ constrain Diophantine exponents, using parametric geometry of numbers and templates to relate properties (A) bad approximability, (B) bounded $X_{i+1}/X_i$, and (C) bounded $L_i/L_{i+1}$. It proves, in particular for the critical case $m=n=2$, that simultaneous boundedness of (B) and (C) yields $\hat{\omega}({\boldsymbol\xi})\le 4/3$, and shows this bound is optimal via explicit constructions; it also develops norm-invariance of the ratio properties and produces extensive template-based constructions realizing prescribed ordinary and uniform exponents. The work blends classical geometry of numbers with Roy’s and David’s variational principle in PGN to map qualitative implications, provide new counterexamples, and address open questions in higher dimensions. It offers a comprehensive framework for understanding how bounded-ratio constraints shape Diophantine exponents, and it lays out concrete constructions and open problems guiding further exploration in multidimensional simultaneous and dual approximation. The results have implications for the structure of best approximations, singularity, and dimension theory in Diophantine approximation on matrices.

Abstract

For a real $m\times n$ matrix $\pmbξ$, we consider its sequence of best Diophantine approximation vectors $ \pmb{x}_i \in \mathbb{Z}^n, \, i =1,2,3, ... $, the sequences of its norms $X_i = \|\pmb{x}_i\|$ and the norms of remainders $L_i = \|\pmbξ\pmb{x}_i\|$. It is known that, in the cases $m=1$, bad approximability of $\pmbξ$ is equivalent to the boundedness of ratios $\frac{X_{i+1}}{X_i}$, while for $n=1$ bad approximability of $\pmbξ$ is equivalent to the boundedness of ratios $ \frac{L_i}{L_{i+1}}$. Moreover, carefully constructed example show that in the cases $m=1$ and $n=1$ boundedness of ratios $ \frac{L_i}{L_{i+1}}$ and $\frac{X_{i+1}}{X_i}$ respectively (the order of ratios changed), does not imply bad approximability of $\pmbξ$. In the present paper, we study the impact of the boundedness of ratios on Diophantine properties of $\pmbξ$, in particular, what restrictions it gives for Diophantine exponents $ω(\pmbξ)$ and $\hatω(\pmbξ)$. One of our particular results deals with the case $m=n=2$. We prove that for $2\times 2 $ matrices $\pmbξ$ boundedness of both ratios $ \frac{X_{i+1}}{X_i}, \frac{L_i}{L_{i+1}} $ implies inequality $\hatω(\pmbξ)\le \frac{4}{3}$ and that this result is optimal. Our methods combine parametric geometry of numbers as well as more classical tools.

Figures (6)

• Figure 1: $h_{\mathbf x_i}(q)$ is figured plain, $h_{\mathbf x_{i+1}}(q)$ is figured dashed, and $h_{{\boldsymbol{\xi}},1}(q)$ is figured thick. Slopes are $-n$ and $m$, here chosen to be $-2$ and $1$.
• Figure 3: Example when $m=3$ and $n=4$. Bold lines represent more than 1 component.
• Figure 4: Construction of $\boldsymbol{P}_{v,w}(p_k,r_k)$ when $\sigma_b > 0$. Slopes different from $-n$ and $m$ are labelled.
• Figure 5: Construction of $\boldsymbol{P}_{v,w}(p_k,r_k)$ when $\sigma_b\leqslant0$. Slopes different from $-n$ and $m$ are labelled.
• Figure 6: Construction of $\boldsymbol{Q}_{v}(q_k,p_k,p_k')$ when $\sigma_b\geqslant 0$. Slopes different from $-n$ and $m$ are labelled.

Theorems & Definitions (38)

• Theorem I: Dirichlet, 1842
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Remark 2.5
• Theorem 2.6
• Remark 2.7
• Theorem 2.8
• Theorem 2.9