JARNIK TYPE THEOREMS ON MANIFOLDS

Hussain M., SCHLEISCHITZ J.

BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, vol.108, no.3, pp.391-405, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 108 Issue: 3
  • Publication Date: 2023
  • Doi Number: 10.1017/s0004972723000291
  • Journal Name: BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.391-405
  • Keywords: Diophantine approximation on manifolds, Jarnik type theorems, Hausdorff measure and dimension, DIOPHANTINE APPROXIMATION, RATIONAL-POINTS, SETS
  • Middle East Technical University Affiliated: Yes

Abstract

Let Psi be a decreasing function. We prove zero-infinity Hausdorff measure criteria for the set of dual Psi-approximable points and for the set of inhomogeneous multiplicative psi-approximable points on nondegenerate planar curves. Our results extend theorems of Huang [Hausdorff theory of dual approximation on planar curves', J. reine angew. Math. 740 (2018), 63-76] and Beresnevich and Velani [A note on three problems in metric Diophantine approximation', in: Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631 (American Mathematical Society, Providence, RI, 2015), 211-229] from s-Hausdorff measure, where s is an element of R, to the more general g-Hausdorff measure, where g is a suitable class of dimension functions.