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Approximation properties of β-expansions II
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Baker, Simon (2018) Approximation properties of β-expansions II. Ergodic Theory and Dynamical Systems, 38 (5). pp. 1627-1641. doi:10.1017/etds.2016.108
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Official URL: https://doi.org/10.1017/etds.2016.108
Abstract
Let be a real number. For a function , define to be the set of such that for infinitely many there exists a sequence satisfying . In Baker [Approximation properties of -expansions. Acta Arith. 168 (2015), 269–287], the author conjectured that for Lebesgue almost every , the condition implies that is of full Lebesgue measure within . In this paper we make a significant step towards proving this conjecture. We prove that given a sequence of positive real numbers satisfying , for Lebesgue almost every , the set is of full Lebesgue measure within . We also study the case where in which the set has Lebesgue measure zero. Applying the mass transference principle developed by Beresnevich and Velani in [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992], we obtain some results on the Hausdorff dimension and the Hausdorff measure of .
| Item Type: | Journal Article | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Diophantine approximation, Number theory | ||||||||
| Journal or Publication Title: | Ergodic Theory and Dynamical Systems | ||||||||
| Publisher: | Cambridge University Press | ||||||||
| ISSN: | 0143-3857 | ||||||||
| Official Date: | August 2018 | ||||||||
| Dates: |
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| Volume: | 38 | ||||||||
| Number: | 5 | ||||||||
| Page Range: | pp. 1627-1641 | ||||||||
| DOI: | 10.1017/etds.2016.108 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access |
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