Going-up theorems for simultaneous Diophantine approximation

Schleischitz J.

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

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 27
  • Basım Tarihi: 2021
  • Dergi Adı: NEW YORK JOURNAL OF MATHEMATICS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.848-880
  • Anahtar Kelimeler: exponents of Diophantine approximation, parametric geometry of numbers, SIMULTANEOUS RATIONAL APPROXIMATION, REAL NUMBER, HAUSDORFF DIMENSION, PARAMETRIC GEOMETRY, UNIFORM EXPONENTS, POINTS, POWERS, SQUARE
  • Orta Doğu Teknik Üniversitesi Adresli: Evet

Özet

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.