%0 Journal Article
%T Diophantine exponents for mildly restricted approximation
%A Yann Bugeaud
%A Simon Kristensen
%J Mathematics
%D 2007
%I arXiv
%R 10.1007/s11512-008-0074-0
%X We are studying the Diophantine exponent \mu_{n,l}$ defined for integers 1 \leq l < n and a vector \alpha \in \mathbb{R}^n by letting \mu_{n,l} = \sup{\mu \geq 0: 0 < ||x \cdot \alpha|| < H(x)^{-\mu} for infinitely many x \in C_{n,l} \cap \mathbb{Z}^n}, where \cdot is the scalar product and || . || denotes the distance to the nearest integer and 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, \infty), with the value n attained for almost all \alpha. We calculate the Hausdorff dimension of the set of vectors \alpha with \mu_{n,l} (\alpha) = \mu for \mu \geq n. Finally, letting w_n denote the exponent obtained by removing the restrictions on x, we show that there are vectors \alpha for which the gaps in the increasing sequence \mu_{n,1} (\alpha) \leq ... \leq \mu_{n,n-1} (\alpha) \leq w_n (\alpha) can be chosen to be arbitrary.
%U http://arxiv.org/abs/0709.0854v1