Simon Kristensen

Diophantine exponents for mildly restricted approximation

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Diophantine exponents for mildly restricted approximation. / Bugeaud, Yann; Kristensen, S.

In: Arkiv foer Matematik, Vol. 47, No. 2, 2009, p. 243-266.

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

Harvard

Bugeaud, Y & Kristensen, S 2009, 'Diophantine exponents for mildly restricted approximation' Arkiv foer Matematik, vol. 47, no. 2, pp. 243-266. DOI: 10.1007/s11512-008-0074-0

APA

CBE

Bugeaud Y, Kristensen S. 2009. Diophantine exponents for mildly restricted approximation. Arkiv foer Matematik. 47(2):243-266. Available from: 10.1007/s11512-008-0074-0

MLA

Bugeaud, Yann and S. Kristensen. "Diophantine exponents for mildly restricted approximation". Arkiv foer Matematik. 2009, 47(2). 243-266. Available: 10.1007/s11512-008-0074-0

Vancouver

Bugeaud Y, Kristensen S. Diophantine exponents for mildly restricted approximation. Arkiv foer Matematik. 2009;47(2):243-266. Available from, DOI: 10.1007/s11512-008-0074-0

Author

Bugeaud, Yann ; Kristensen, S./ Diophantine exponents for mildly restricted approximation. In: Arkiv foer Matematik. 2009 ; Vol. 47, No. 2. pp. 243-266

Bibtex

@article{9a161280172911dfb95d000ea68e967b,
title = "Diophantine exponents for mildly restricted approximation",
abstract = "We are studying the Diophantine exponent μ n,l defined for integers 1≤l<n and a vector α∈ℝ n by letting where is the scalar product, denotes the distance to the nearest integer and is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the exponent takes all values in the interval [l+1,∞), with the value n attained for almost all α. We calculate the Hausdorff dimension of the set of vectors α with μ n,l (α)=μ for μ≥n. Finally, letting w n denote the exponent obtained by removing the restrictions on , we show that there are vectors α for which the gaps in the increasing sequence μ n,1(α)≤...≤μ n,n-1(α)≤w n (α) can be chosen to be arbitrary.",
author = "Yann Bugeaud and S. Kristensen",
year = "2009",
doi = "10.1007/s11512-008-0074-0",
language = "English",
volume = "47",
pages = "243--266",
journal = "Arkiv foer Matematik",
issn = "0004-2080",
publisher = "Springer Netherlands",
number = "2",

}

RIS

TY - JOUR

T1 - Diophantine exponents for mildly restricted approximation

AU - Bugeaud,Yann

AU - Kristensen,S.

PY - 2009

Y1 - 2009

N2 - We are studying the Diophantine exponent μ n,l defined for integers 1≤l<n and a vector α∈ℝ n by letting where is the scalar product, denotes the distance to the nearest integer and is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the exponent takes all values in the interval [l+1,∞), with the value n attained for almost all α. We calculate the Hausdorff dimension of the set of vectors α with μ n,l (α)=μ for μ≥n. Finally, letting w n denote the exponent obtained by removing the restrictions on , we show that there are vectors α for which the gaps in the increasing sequence μ n,1(α)≤...≤μ n,n-1(α)≤w n (α) can be chosen to be arbitrary.

AB - We are studying the Diophantine exponent μ n,l defined for integers 1≤l<n and a vector α∈ℝ n by letting where is the scalar product, denotes the distance to the nearest integer and is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the exponent takes all values in the interval [l+1,∞), with the value n attained for almost all α. We calculate the Hausdorff dimension of the set of vectors α with μ n,l (α)=μ for μ≥n. Finally, letting w n denote the exponent obtained by removing the restrictions on , we show that there are vectors α for which the gaps in the increasing sequence μ n,1(α)≤...≤μ n,n-1(α)≤w n (α) can be chosen to be arbitrary.

U2 - 10.1007/s11512-008-0074-0

DO - 10.1007/s11512-008-0074-0

M3 - Journal article

VL - 47

SP - 243

EP - 266

JO - Arkiv foer Matematik

T2 - Arkiv foer Matematik

JF - Arkiv foer Matematik

SN - 0004-2080

IS - 2

ER -