Exact approximation order and well-distributed sets

We prove that for a suitable Ahlfors regular metric measure space X and a function ψ:(0,∞)→(0,∞) from a suitable class of approximation functions, the Hausdorff dimensions of the set Wψ(Q) of all points ψ-well-approximable by a well-distributed subset Q⊂X, and the set Eψ(Q) of points that are exactl...

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Bibliographic Details
Published inAdvances in mathematics (New York. 1965) Vol. 414; p. 108871
Main Authors Bandi, Prasuna, Ghosh, Anish, Nandi, Debanjan
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.02.2023
Subjects
Diophantine approximation
Hausdorff dimension
Negatively curved spaces
Well-distributed systems
secondary
Diophantine approximation
Hausdorff dimension
Well-distributed systems
Negatively curved spaces
primary
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Summary:We prove that for a suitable Ahlfors regular metric measure space X and a function ψ:(0,∞)→(0,∞) from a suitable class of approximation functions, the Hausdorff dimensions of the set Wψ(Q) of all points ψ-well-approximable by a well-distributed subset Q⊂X, and the set Eψ(Q) of points that are exactly ψ-approximable by Q, coincide. This answers in a general setting, a question of Beresnevich-Dickinson-Velani in the case of approximation of reals by rationals, and answered by Bugeaud in that case using the continued-fraction expansion of reals. Our main result applies in particular to approximation by orbits of fixed points of a wide class of discrete groups of isometries acting on the boundary of hyperbolic metric spaces.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2023.108871