Uniform Random Covering Problems
(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
- Koivusalo, Henna ; Liao, Lingmin and Persson, Tomas LU
- organization
- publishing date
- 2023-01-01
- 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 &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 ∈ &x1D54B;: ∀N ≫ 1, ∃n ≤ N, s.t. |ω<sub>n</sub> - y| < 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}}, }