A FrostmanType Lemma for Sets with Large Intersections, and an Application to Diophantine Approximation
(2015) In Proceedings of the Edinburgh Mathematical Society 58(2). p.521542 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 Frostmantype 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 En. 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 Frostmantype 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 En. 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 Elambda(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
 14643839
 DOI
 10.1017/S0013091514000066
 language
 English
 LU publication?
 yes
 id
 55e9f0fa2246464f8dade66b76e37691 (old id 7432960)
 alternative location
 https://arxiv.org/abs/1302.0954
 date added to LUP
 20160401 13:06:29
 date last changed
 20230718 09:21:01
@article{55e9f0fa2246464f8dade66b76e37691, 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 Frostmantype 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 En. 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 Elambda(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 = {{14643839}}, keywords = {{classes with large intersections; diophantine approximation; Hausdorff dimension}}, language = {{eng}}, number = {{2}}, pages = {{521542}}, publisher = {{Cambridge University Press}}, series = {{Proceedings of the Edinburgh Mathematical Society}}, title = {{A FrostmanType 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}}, }