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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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 > 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:	Date Event August 2018 Published 14 February 2017 Available 4 September 2016 Accepted
Date of first compliant deposit:	16 September 2016
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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