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.