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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)
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author
organization
publishing date
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)
date added to LUP
2015-06-24 13:35:25
date last changed
2017-03-14 13:40:55
@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) &lt; 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 &gt; 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) &lt; 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 &lt;= 1/alpha. This improves one of our previously published results.},
  author       = {Persson, Tomas and Reeve, Henry W. J.},
  issn         = {1464-3839},
  keyword      = {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},
  volume       = {58},
  year         = {2015},
}