On geometry of numbers and uniform rational approximation to the Veronese curve


Schleischitz J.

Periodica Mathematica Hungarica, 2021 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume:
  • Publication Date: 2021
  • Doi Number: 10.1007/s10998-021-00382-1
  • Title of Journal : Periodica Mathematica Hungarica
  • Keywords: Exponents of Diophantine approximation, Regular graph, Parametric geometry of numbers, DIOPHANTINE APPROXIMATION, PARAMETRIC GEOMETRY, REAL NUMBER, EXPONENTS, SCHMIDT

Abstract

© 2021, Akadémiai Kiadó, Budapest, Hungary.Consider the classical problem of rational simultaneous approximation to a point in Rn. The optimal lower bound on the gap between the induced ordinary and uniform approximation exponents has been established by Marnat and Moshchevitin in 2018. Recently Nguyen, Poels and Roy provided information on the best approximating rational vectors to the points where the gap is close to this minimal value. Combining the latter result with parametric geometry of numbers, we effectively bound the dual linear form exponents in the described situation. As an application, we slightly improve the upper bound for the classical exponent of uniform Diophantine approximation λ^ n(ξ) , for even n≥ 4. Unfortunately our improvements are small, for n= 4 only in the fifth decimal digit. However, the underlying method in principle can be improved with more effort to provide better bounds. We indeed establish reasonably stronger results for numbers which almost satisfy equality in the estimate by Marnat and Moshchevitin. We conclude with consequences on the classical problem of approximation to real numbers by algebraic numbers/integers of uniformly bounded degree.