  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.
