The mixed Schmidt conjecture in the theory of Diophantine approximation.

Let xs1D49F=(dn)∞n=1 be a sequence of integers with dn≥2, and let (i,j) be a pair of strictly positive numbers with i+j=1. We prove that the set of xxs2208xs211D for which there exists some constant c(x)≧0 such that \[ \max \!\big \{|q|_\mathcal {D}^{1/i}, \|qx\|^{1/j}\big \} > c(x)/ q \quad \mbox {for all } q \in \mathbb {N} \] is one-quarter winning (in the sense of Schmidt games). Thus the intersection of any countable number of such sets is of full dimension. This, in turn, establishes the natural analogue of Schmidt’s conjecture within the framework of the de Mathan–Teulié conjecture, also known as the “mixed Littlewood conjecture”