Exact uniform approximation and Dirichlet spectrum in dimension at least two

• Johannes Schleischitz ^[1]
  1. [1] Middle East Technical University

     Middle East Technical University

     Turquía

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 29, Nº. 5, 2023
• Idioma: inglés
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• Resumen

  • For m ≥ 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 proved existence of such vectors but without being able to provide any concrete value in the Dirichlet spectrum. Our constructive proof is considerably shorter and less involved than previous work on the topic. Moreover, it is flexible enough to show that the according set of vectors with prescribed Dirichlet constant has large packing dimension and rather large Hausdorff dimension as well, thereby contributing to the metrical problem raised in the aforementioned paper by Beresnevich et alia. We further establish a more general result on exact uniform approximation, applicable to a wide class of approximating functions. Moreover, minor twists in the proof yield similar, slightly weaker results when restricting to a certain class of classical fractals or considering other norms on Rm. In an Appendix we address the situation of a linear form.