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

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

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

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

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

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

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

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

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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",

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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 -