Let $\mathcal{D}=(d_n)_{n=1}^\infty$ be a bounded sequence of integers with
$d_n\ge 2$ and let $(i, j)$ be a pair of strictly positive numbers with
$i+j=1$. We prove that the set of $x \in \RR$ for which there exists some
constant $c(x) > 0$ such that \[ \max\{|q|_\DDD^{1/i}, \|qx\|^{1/j}\} > c(x)/ q
\qquad \forall q \in \NN \] is one quarter winning (in the sense of Schmidt
games). Thus the intersection of any countable number of such sets is of full
dimension.