We are studying the Diophantine exponent μ

_{ n,l } defined for integers 1≤

*l*<

*n* and a vector α∈ℝ

^{ n } by letting

where

is the scalar product,

denotes the distance to the nearest integer and

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 μ

_{ n,l }(α)=μ for μ≥

*n*. Finally, letting

*w* _{ n } denote the exponent obtained by removing the restrictions on

, we show that there are vectors α for which the gaps in the increasing sequence μ

_{ n,1}(α)≤...≤μ

_{ n,n-1}(α)≤

*w* _{ n }(α) can be chosen to be arbitrary.