Going-up theorems for simultaneous Diophantine approximation

Schleischitz J.

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

  • Publication Type: Article / Article
  • Volume: 27
  • Publication Date: 2021
  • Journal Name: NEW YORK JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • 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
  • Middle East Technical University Affiliated: Yes

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.