The mixed Schmidt conjecture in the theory of Diophantine approximation

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. In turn, this establishes the natural analogue of Schmidt's conjecture within the framework of the de Mathan-Teuli\'e conjecture -- also known as the `Mixed Littlewood Conjecture'.