论文标题
确切的均匀近似和dirichlet频谱至少两个
Exact uniform approximation and Dirichlet spectrum in dimension at least two
论文作者
论文摘要
对于$ m \ geq 2 $，我们确定$ \ rm $的dirichlet频谱相对于同时近似和最大范围作为整个间隔$ [0,1] $。这补充了几位作者的先前工作，尤其是Akhunzhanov和Moshchevitin，他们认为$ M = 2 $和Euclidean Norm。我们构建了真正的liouville载体的明确示例，这些载体意识到单位间隔中的任何值。特别是，对于正值，它们既不是近似也不是奇异的。因此，在最近的一篇论文中，我们获得了贝雷斯内维奇，关，玛娜特，拉米雷斯和维拉尼的主要主张的建设性证明，他们在间隔中获得了$ [0,1] $的可数分区，每个分区都与Dirichlet Spectrum具有非空交点。我们的构造足够灵活，可以表明带有规定的Dirichlet指数的向量集具有较大的包装尺寸，并且还具有较大的Hausdorff尺寸。我们在精确均匀的近似值上建立了更一般的结果，适用于一类近似近似功能。我们的建设性证明比以前在主题上的工作要短得多，而且参与其中。通过对证据的微小扭曲，我们推断出限制到某种类别的经典分形或其他规范时相似的，稍弱的结果。在附录中，我们解决了线性形式的情况。
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, Ramírez 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.
