We are studying the Diophantine exponent defined for integers and a vector by letting , where is the scalar product and denotes the distance to the nearest integer and is the generalised cone consisting of all vectors with the height attained among the first coordinates. We show that the exponent takes all values in the interval , with the value attained for almost all . We calculate the Hausdorff dimension of the set of vectors with for . Finally, letting denote the exponent obtained by removing the restrictions on , we show that there are vectors for which the gaps in the increasing sequence can be chosen to be arbitrary.