Arkiv för Matematik

  • Ark. Mat.
  • Volume 47, Number 2 (2009), 243-266.

Diophantine exponents for mildly restricted approximation

Yann Bugeaud and Simon Kristensen

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We are studying the Diophantine exponent μn, l defined for integers 1≤l< n and a vector α∈ℝn by letting $\mu_{n,l}=\sup\{\mu\geq0: 0 < \Vert\underline{x}\cdot\alpha\Vert<H(\underline{x})^{-\mu}\ \text{for infinitely many}\ \underline{x}\in\mathcal{C}_{n,l}\cap\mathbb{Z}^n\},$ where $\cdot$ is the scalar product, $\|\cdot\|$ denotes the distance to the nearest integer and $\mathcal{C}_{n,l}$ 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 μn, l(α)=μ for μ≥n. Finally, letting wn denote the exponent obtained by removing the restrictions on $\underline{x}$, we show that there are vectors α for which the gaps in the increasing sequence μn,1(α)≤...≤μn, n-1(α)≤wn(α) can be chosen to be arbitrary.

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Ark. Mat., Volume 47, Number 2 (2009), 243-266.

Received: 10 September 2007
First available in Project Euclid: 31 January 2017

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Bugeaud, Yann; Kristensen, Simon. Diophantine exponents for mildly restricted approximation. Ark. Mat. 47 (2009), no. 2, 243--266. doi:10.1007/s11512-008-0074-0.

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