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Diophantine exponents for mildly restricted approximation

By Yann Bugeaud and Simon Kristensen
No. 09, September 2007
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 $\mu_{n,\ell} = \sup\{\mu \geq 0: 0 < | \underline{x} \cdot \alpha| < H(\underline{x})^{-\mu} \text{ for infinitely many } \underline{x} \in \mathcal{C}_{n,\ell} \cap \mathbb{Z}^n \}$, 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.
Published in Arkiv för Matematik, Volume 47, Number 2 / October, 2009, pp 243-266.
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