  For any $i,j \ge 0$ with $i+j =1$, let $\bad(i,j)$ denote the set of points
$(x,y) \in \R^2$ for which $ \max \{\|qx\|^{1/i}, \|qy\|^{1/j} \} > c/q $ for
all $ q \in \N $. Here $c = c(x,y)$ is a positive constant. Our main result
implies that any finite intersection of such sets has full dimension. This
settles a conjecture of Wolfgang M. Schmidt in the theory of simultaneous
Diophantine approximation.
