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Uniform Random Covering Problems

Koivusalo, Henna ; Liao, Lingmin and Persson, Tomas LU orcid (2023) In International Mathematics Research Notices 2023(1). p.455-481
Abstract

Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence ω = (ωn)n≥1 uniformly distributed on the unit circle &x1D54B; and a sequence (rn)n≥1 of positive real numbers with limit 0. We investigate the size of the random set U(ω) := {y ∈ &x1D54B;: ∀N ≫ 1, ∃n ≤ N, s.t. |ωn - y| < rN}. Some sufficient conditions for U(ω) to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that U(ω) is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff... (More)

Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence ω = (ωn)n≥1 uniformly distributed on the unit circle &x1D54B; and a sequence (rn)n≥1 of positive real numbers with limit 0. We investigate the size of the random set U(ω) := {y ∈ &x1D54B;: ∀N ≫ 1, ∃n ≤ N, s.t. |ωn - y| < rN}. Some sufficient conditions for U(ω) to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that U(ω) is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.

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author
; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
International Mathematics Research Notices
volume
2023
issue
1
pages
27 pages
publisher
Oxford University Press
external identifiers
  • scopus:85139893854
ISSN
1073-7928
DOI
10.1093/imrn/rnab272
language
English
LU publication?
yes
additional info
Publisher Copyright: © The Author(s) 2021. Published by Oxford University Press.
id
d9825afe-a88a-4a7a-9206-01ec4fa4e068
date added to LUP
2023-06-16 22:52:09
date last changed
2023-08-21 16:36:19
@article{d9825afe-a88a-4a7a-9206-01ec4fa4e068,
  abstract     = {{<p>Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence ω = (ω<sub>n</sub>)<sub>n≥1</sub> uniformly distributed on the unit circle &amp;x1D54B; and a sequence (r<sub>n</sub>)<sub>n≥1</sub> of positive real numbers with limit 0. We investigate the size of the random set U(ω) := {y ∈ &amp;x1D54B;: ∀N ≫ 1, ∃n ≤ N, s.t. |ω<sub>n</sub> - y| &lt; r<sub>N</sub>}. Some sufficient conditions for U(ω) to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that U(ω) is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.</p>}},
  author       = {{Koivusalo, Henna and Liao, Lingmin and Persson, Tomas}},
  issn         = {{1073-7928}},
  language     = {{eng}},
  month        = {{01}},
  number       = {{1}},
  pages        = {{455--481}},
  publisher    = {{Oxford University Press}},
  series       = {{International Mathematics Research Notices}},
  title        = {{Uniform Random Covering Problems}},
  url          = {{http://dx.doi.org/10.1093/imrn/rnab272}},
  doi          = {{10.1093/imrn/rnab272}},
  volume       = {{2023}},
  year         = {{2023}},
}