Going-up theorems for simultaneous Diophantine approximation


Schleischitz J.

NEW YORK JOURNAL OF MATHEMATICS, vol.27, pp.848-880, 2021 (Journal Indexed in SCI) identifier

  • Publication Type: Article / Article
  • Volume: 27
  • Publication Date: 2021
  • Title of Journal : NEW YORK JOURNAL OF MATHEMATICS
  • Page Numbers: pp.848-880
  • Keywords: exponents of Diophantine approximation, parametric geometry of numbers, SIMULTANEOUS RATIONAL APPROXIMATION, REAL NUMBER, HAUSDORFF DIMENSION, PARAMETRIC GEOMETRY, UNIFORM EXPONENTS, POINTS, POWERS, SQUARE

Abstract

We establish several new inequalities linking classical exponents of Diophantine approximation associated to a real vector (xi) under bar = (xi, xi(2), ..., xi(N)) in various dimensions N. We thereby obtain variants, and partly refinements, of recent results of Badziahin and Bugeaud. We further implicitly recover inequalities of Bugeaud and Laurent as special cases, with new proofs. Similar estimates concerning general real vectors (not on the Veronese curve) with Q-linearly independent coordinates are addressed as well. Our method is based on Minkowski's Second Convex Body Theorem, applied in the framework of parametric geometry of numbers introduced by Schmidt and Summerer. We also frequently employ Mahler's Duality result on polar convex bodies.