Diophantine exponents for mildly restricted approximation

Yann Bugeaud; Simon Kristensen

Diophantine exponents for mildly restricted approximation

Research output: Working paper/Preprint › Working paper › Research

Abstract

We are studying the Diophantine exponent $\mu_{n,\ell}$ defined for integers $1 \leq \ell < n$ and a vector $\alpha \in \mathbb{R}^n$ by letting $\mu_{n,\ell} = \sup\{\mu \geq 0: 0 < | \underline{x} \cdot \alpha| < H(\ux)^{-\mu} \text{ for infinitely many } \underline{x} \in \mathcal{C}_{n,\ell} \cap \mathbb{Z}^n \}$ , where $\cdot$ is the scalar product and $| \cdot |$ denotes the distance to the nearest integer and $\mathcal{C}_{n,\ell}$ is the generalised cone consisting of all vectors with the height attained among the first $\ell$ coordinates. We show that the exponent takes all values in the interval $[\ell+1, \infty)$ , with the value $n$ attained for almost all $\alpha$ . We calculate the Hausdorff dimension of the set of vectors $\alpha$ with $\mu_{n,\ell} (\alpha) = \mu$ for $\mu \geq n$ . Finally, letting $w_n$ denote the exponent obtained by removing the restrictions on $\underline{x}$ , we show that there are vectors $\alpha$ for which the gaps in the increasing sequence $\mu_{n,1} (\alpha) \leq \cdots \leq \mu_{n,n-1} (\alpha) \leq w_n (\alpha)$ can be chosen to be arbitrary.

Original language	English
Place of publication	Århus
Publisher	Department of Mathematical Sciences , University of Aarhus
Number of pages	21
Publication status	Published - 2007

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title = "Diophantine exponents for mildly restricted approximation",

abstract = "We are studying the Diophantine exponent defined for integers and a vector by letting , where is the scalar product and denotes the distance to the nearest integer and is the generalised cone consisting of all vectors with the height attained among the first coordinates. We show that the exponent takes all values in the interval , with the value attained for almost all . We calculate the Hausdorff dimension of the set of vectors with for . Finally, letting denote the exponent obtained by removing the restrictions on , we show that there are vectors for which the gaps in the increasing sequence can be chosen to be arbitrary.",

author = "Yann Bugeaud and Simon Kristensen",

year = "2007",

language = "English",

publisher = "Department of Mathematical Sciences , University of Aarhus",

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T1 - Diophantine exponents for mildly restricted approximation

AU - Bugeaud, Yann

AU - Kristensen, Simon

PY - 2007

Y1 - 2007

N2 - We are studying the Diophantine exponent defined for integers and a vector by letting , where is the scalar product and denotes the distance to the nearest integer and is the generalised cone consisting of all vectors with the height attained among the first coordinates. We show that the exponent takes all values in the interval , with the value attained for almost all . We calculate the Hausdorff dimension of the set of vectors with for . Finally, letting denote the exponent obtained by removing the restrictions on , we show that there are vectors for which the gaps in the increasing sequence can be chosen to be arbitrary.

AB - We are studying the Diophantine exponent defined for integers and a vector by letting , where is the scalar product and denotes the distance to the nearest integer and is the generalised cone consisting of all vectors with the height attained among the first coordinates. We show that the exponent takes all values in the interval , with the value attained for almost all . We calculate the Hausdorff dimension of the set of vectors with for . Finally, letting denote the exponent obtained by removing the restrictions on , we show that there are vectors for which the gaps in the increasing sequence can be chosen to be arbitrary.

M3 - Working paper

BT - Diophantine exponents for mildly restricted approximation

PB - Department of Mathematical Sciences , University of Aarhus

CY - Århus

ER -

Diophantine exponents for mildly restricted approximation

Abstract

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