The Library
An analogue of Khintchine's theorem for self-conformal sets
Tools
Baker, Simon (2019) An analogue of Khintchine's theorem for self-conformal sets. Mathematical Proceedings of the Cambridge Philosophical Society, 167 (3). pp. 567-597. doi:10.1017/S030500411800049X ISSN 0305-0041.
|
PDF
WRAP-analogue-Khintchines-therorem-self-conformal-sets-Baker-2018.pdf - Accepted Version - Requires a PDF viewer. Download (743Kb) | Preview |
Official URL: https://doi.org/10.1017/S030500411800049X
Abstract
Khintchine's theorem is a classical result from metric number theory which relates the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. In this paper we ask whether an analogous result holds for iterated function systems (IFS's). We say that an IFS is approximation regular if we observe Khintchine type behaviour, i.e., if the size of certain limsup sets defined using the IFS is determined by the convergence/divergence of naturally occurring sums. We prove that an IFS is approximation regular if it consists of conformal mappings and satisfies the strong separation condition, or if it is a collection of similarities and satisfies the open set condition. The divergence condition we introduce incorporates the inhomogeneity present within the IFS. We demonstrate via an example that such an approach is essential. We also formulate an analogue of the Duffin-Schaeffer conjecture and show that it holds for a set of full Hausdorff dimension.
Combining our results with the mass transference principle of Beresnevich and Velani \cite{BerVel}, we prove a general result that implies the existence of exceptional points within the attractor of our IFS. These points are exceptional in the sense that they are "very well approximated". As a corollary of this result, we obtain a general solution to a problem of Mahler, and prove that there are badly approximable numbers that are very well approximated by quadratic irrationals.
The ideas put forward in this paper are introduced in the general setting of IFS's that may contain overlaps. We believe that by viewing IFS's from the perspective of metric number theory, one can gain a greater insight into the extent to which they overlap. The results of this paper should be interpreted as a first step in this investigation.
Item Type: | Journal Article | ||||||||
---|---|---|---|---|---|---|---|---|---|
Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Number theory, Measure theory, Hausdorff measures | ||||||||
Journal or Publication Title: | Mathematical Proceedings of the Cambridge Philosophical Society | ||||||||
Publisher: | Cambridge University Press | ||||||||
ISSN: | 0305-0041 | ||||||||
Official Date: | November 2019 | ||||||||
Dates: |
|
||||||||
Volume: | 167 | ||||||||
Number: | 3 | ||||||||
Page Range: | pp. 567-597 | ||||||||
DOI: | 10.1017/S030500411800049X | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Re-use Statement: | This article has been published in a revised form in Mathematical Proceedings of the Cambridge Philosophical Society https://doi.org/10.1017/S030500411800049X. 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. © copyright holder. | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Copyright Holders: | Cambridge Philosophical Society 2018 | ||||||||
Date of first compliant deposit: | 13 June 2018 | ||||||||
Date of first compliant Open Access: | 4 February 2019 | ||||||||
Related URLs: | |||||||||
Open Access Version: |
Request changes or add full text files to a record
Repository staff actions (login required)
View Item |
Downloads
Downloads per month over past year