- Title
- A dichotomy law for the diophantine properties in β-dynamical systems
- Creator
- Coons, Michael; Hussain, Mumtaz; Wang, Bao-Wei
- Relation
- ARC.DE140100223 http://purl.org/au-research/grants/arc/DE140100223
- Relation
- Mathematika Vol. 62, Issue 3, p. 884-897
- Publisher Link
- http://dx.doi.org/10.1112/S0025579316000085
- Publisher
- London Mathematical Society
- Resource Type
- journal article
- Date
- 2016
- Description
- Let β> 1 be a real number and define the β-transformation on [0,1] by Tβ : x ↦ βx mod 1. Further, define: [formula could not be replicated] and [formula could not be replicated], where Ψ : ℕ → ℝ>0 is a positive function such that Ψ(n) → 0 as n → ∞. In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalized Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.
- Subject
- dichotomy law; diophantine properties; β-dynamical systems
- Identifier
- http://hdl.handle.net/1959.13/1323831
- Identifier
- uon:24900
- Identifier
- ISSN:0025-5793
- Rights
- This article has been published in a revised form in Mathematika http://dx.doi.org/10.1112/S0025579316000085. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © University College London.
- Language
- eng
- Full Text
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