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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. 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