The set of $Φ$ badly approximable matrices has full Hausdorff dimension

TL;DR

The paper advances the theory of simultaneous Diophantine approximation for real matrices by extending the Hausdorff and packing-dimension results for the sets of $\Phi$-badly approximable matrices from the power-law case to general decreasing $\Phi$, under mild regularity (monotonicity or condition (C1)). Using a uniform version of the Das–Fishman–Simmons–Urbański variational principle applied to parametric geometry of numbers and template theory, it constructs templates whose associated trajectories enforce bad-approximation behavior and yields exact dimension formulas: $\dim_H(Bad(\Phi))=\dim_H(W_{\tau})=(n-1)m+(m+n)/(1+\tau)$ and $\dim_P(Bad(\Phi))=\dim_P(W_{\tau})=mn$, with analogous results for the smaller set $Exact(\Phi)$ under stronger assumptions. The results hold in the general matrix setting, and the paper provides new lower bounds for $Exact(\Phi)$ and corresponding packing-dimension statements, including for the Liouville regime $\tau=\infty$ where full dimension is obtained under certain hypotheses. The work connects to recent refinements by Bandi and de Saxcé and complements existing results by Ward and others, broadening the scope of known dimension phenomena in simultaneous Diophantine approximation. Overall, the paper provides a robust framework to transfer dimension results from power-law to general $\Phi$ via template-based variational methods, with potential implications for related problems in multiplicative and additive Diophantine approximation.

Abstract

Recently Koivusalo, Levesley, Ward and Zhang introduced the set of simultaneously $Φ$-badly approximable real vectors of $\mathbb{R}^m$ with respect to an approximation function $Φ$, and determined its Hausdorff dimension for the special class of power functions $Φ(t)=t^{-τ}$. We refine this by naturally extending the formula to arbitrary decreasing functions, in terms of the lower order of $1/Φ$ at infinity. We also provide an alternative, rather mild condition on $Φ$ for this conclusion. Moreover, our results apply in the general matrix setting, and we establish an according formula for packing dimension as well. Thereby we also complement a recent refinement by Bandi and de Saxcé on the smaller set of exact approximation with respect to $Φ$. Our basic tool is (a uniform variant of) the variational principle by Das, Fishman, Simmons, Urbański. We also prove some new lower estimates regarding the set of exact approximation order in the matrix setting, which are sharp in special instances. For this we combine the result by Bandi and de Saxcé with a method developed by Moshchevitin.

Figures (1)

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Theorems & Definitions (19)

• Conjecture 1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 1
• Definition 1
• Theorem 3.1: Das, Fishman, Simmons, Urbański
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3