Exact uniform approximation and Dirichlet spectrum in dimension at least two

TL;DR

This paper fully characterizes the Dirichlet spectrum in higher dimensions, constructing explicit vectors with prescribed approximation properties, and demonstrates the large size of sets with specific Dirichlet exponents.

Contribution

It provides a constructive, simplified proof of the Dirichlet spectrum in higher dimensions and extends results to various norms, fractal sets, and linear forms.

Findings

The Dirichlet spectrum in $ m extbf{R}^m$ is the entire interval [0,1].

Explicit Liouville vectors can realize any value in [0,1].

Sets with prescribed Dirichlet exponents have large Hausdorff and packing dimensions.

Abstract

For $m\geq 2$, we determine the Dirichlet spectrum in $\Rm$ with respect to simultaneous approximation and the maximum norm as the entire interval $[0,1]$. This complements previous work of several authors, especially Akhunzhanov and Moshchevitin, who considered $m=2$ and Euclidean norm. We construct explicit examples of real Liouville vectors realizing any value in the unit interval. In particular, for positive values, they are neither badly approximable nor singular. Thereby we obtain a constructive proof of the main claim in a recent paper by Beresnevich, Guan, Marnat, Ram\'irez and Velani, who obtained a countable partition of $[0,1]$ into intervals with each having non-empty intersection with the Dirichlet spectrum. Our construction is flexible enough to show that the according set of vectors with prescribed Dirichlet exponent has large packing dimension and rather large Hausdorff dimension as well. We establish a more general result on exact uniform approximation, applicable to a wide class of approximating functions. Our constructive proofs are considerably shorter and less involved than previous work on the topic. By minor twists of our proof, we infer similar, slightly weaker results when restricting to a certain class of classical fractal sets or other norms. In an Appendix we address the situation of a linear form.