A Frostman-Type Lemma for Sets with Large Intersections, and an Application to Diophantine Approximation
Persson, Tomas LU
and Reeve, Henry W. J.
(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}},
}