A number a is called badly approximable if j a-p/q | > c/q2 for some c > 0 and all rationals pjq. It is known that an irrational number a is badly approximable if and only if the partial denominators in its continued fraction are bounded [4, Theorem 23]. In a recent paper [7] I proved results of the following type: // fuf2,•■• are differenliable functions whose derivatives are continuous and vanish nowhere, then there are continuum-many numbers a such that all the numbers fi(a),f2(a), ••• are badly approximable. Let 0 which assigns to every Be£l a nonempty set (B) <= Í2 such that for C e &(B), «(C) c a(B).

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Let K be a field of characteristic 0 and let n be a natural number. Let Γ be a subgroup of the multiplicative group (K *

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