A Frostman-Type Lemma for Sets with Large Intersections, and an Application to Diophantine Approximation
(2015) In Proceedings of the Edinburgh Mathematical Society 58(2). p.521-542- Abstract
- We consider classes G(s)([0, 1]) of subsets of [0, 1], originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least s. We provide a Frostman-type lemma to determine if a limsup set is in such a class. Suppose that E = lim supE(n) subset of [0, 1], and that mu(n) are probability measures with support in E-n. If there exists a constant C such that integral integral vertical bar x - y vertical bar(-s) d mu(n)(x) d mu(n)(y) < C for all n, then, under suitable conditions on the limit measure of the sequence (mu(n)), we prove that the set E is in the class G(s)([0, 1]). As an application we prove that, for alpha > 1 and almost all lambda is an... (More)
- We consider classes G(s)([0, 1]) of subsets of [0, 1], originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least s. We provide a Frostman-type lemma to determine if a limsup set is in such a class. Suppose that E = lim supE(n) subset of [0, 1], and that mu(n) are probability measures with support in E-n. If there exists a constant C such that integral integral vertical bar x - y vertical bar(-s) d mu(n)(x) d mu(n)(y) < C for all n, then, under suitable conditions on the limit measure of the sequence (mu(n)), we prove that the set E is in the class G(s)([0, 1]). As an application we prove that, for alpha > 1 and almost all lambda is an element of (1/2, 1), the set E-lambda(alpha) = {x is an element of [0, 1] : vertical bar x - s(n vertical bar) < 2(-alpha n) infinitely often}, where s(n) is an element of {(1 - lambda)Sigma(n)(k=0) a(k)lambda(k) and a(k) is an element of {0, 1}}, belongs to the class G(s) for s <= 1/alpha. This improves one of our previously published results. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/7432960
- author
- Persson, Tomas LU and Reeve, Henry W. J.
- organization
- publishing date
- 2015
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- classes with large intersections, diophantine approximation, Hausdorff dimension
- in
- Proceedings of the Edinburgh Mathematical Society
- volume
- 58
- issue
- 2
- pages
- 521 - 542
- publisher
- Cambridge University Press
- external identifiers
-
- wos:000353900900014
- scopus:84928594501
- ISSN
- 1464-3839
- DOI
- 10.1017/S0013091514000066
- language
- English
- LU publication?
- yes
- id
- 55e9f0fa-2246-464f-8dad-e66b76e37691 (old id 7432960)
- alternative location
- https://arxiv.org/abs/1302.0954
- date added to LUP
- 2016-04-01 13:06:29
- date last changed
- 2023-07-18 09:21:01
@article{55e9f0fa-2246-464f-8dad-e66b76e37691, abstract = {{We consider classes G(s)([0, 1]) of subsets of [0, 1], originally introduced by Falconer, that are closed under countable intersections, and such that every set in the class has Hausdorff dimension at least s. We provide a Frostman-type lemma to determine if a limsup set is in such a class. Suppose that E = lim supE(n) subset of [0, 1], and that mu(n) are probability measures with support in E-n. If there exists a constant C such that integral integral vertical bar x - y vertical bar(-s) d mu(n)(x) d mu(n)(y) < C for all n, then, under suitable conditions on the limit measure of the sequence (mu(n)), we prove that the set E is in the class G(s)([0, 1]). As an application we prove that, for alpha > 1 and almost all lambda is an element of (1/2, 1), the set E-lambda(alpha) = {x is an element of [0, 1] : vertical bar x - s(n vertical bar) < 2(-alpha n) infinitely often}, where s(n) is an element of {(1 - lambda)Sigma(n)(k=0) a(k)lambda(k) and a(k) is an element of {0, 1}}, belongs to the class G(s) for s <= 1/alpha. This improves one of our previously published results.}}, author = {{Persson, Tomas and Reeve, Henry W. J.}}, issn = {{1464-3839}}, keywords = {{classes with large intersections; diophantine approximation; Hausdorff dimension}}, language = {{eng}}, number = {{2}}, pages = {{521--542}}, publisher = {{Cambridge University Press}}, series = {{Proceedings of the Edinburgh Mathematical Society}}, title = {{A Frostman-Type Lemma for Sets with Large Intersections, and an Application to Diophantine Approximation}}, url = {{http://dx.doi.org/10.1017/S0013091514000066}}, doi = {{10.1017/S0013091514000066}}, volume = {{58}}, year = {{2015}}, }