Spectrum of the exponents of best rational approximation

Abstract

Using the new theory of W. M. Schmidt and L. Summerer called parametric geometry of numbers, we show that the going-up and going-down transference inequalities of W. M. Schmidt and M. Laurent describe the full spectrum of the $n$ exponents of best rational approximation to points in ${\mathbb{R}}^{n+1}$.

13 pages, 4 figures, minor corrections since version 1

References in corpus (1)

• On Schmidt and Summerer parametric geometry of numbers

Cited by in corpus (9)

• A variational principle in the parametric geometry of numbers
• A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation
• On the topology of Diophantine approximation Spectra
• Diophantine sets and Dirichlet improvability
• Exponents of Diophantine approximation in dimension two for a general class of numbers
• Geometry of Diophantine exponents
• The complex case of Schmidt's going-down Theorem
• On connectedness in the parametric geometry of numbers
• Independence of the Diophantine exponents associated with linear subspaces