On Badly Approximable Numbers and Certain Games

1. Introduction. 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<a<l/2, 0 < ß < 1, and consider the following game of two players black and white. First black picks a closed interval B¡. Then white picks a closed interval Wx c By whose length is a times the length of Bt. Then black chooses an interval B2 <= Wx which is closed and has length ß times the length of IF,. Then again white picks a closed interval W2 c B2 of length a times the length of B2, and so on. Call white the winner of a play if the intersection of the intervals W} is badly approximable; otherwise black is called the winner. Who will win? Since the badly approximable numbers have Lebesgue measure zero, [4, Theorem 29], one might think that black can always win. It turns out, however, that white can always win (Theorem 3). We shall show that sets S with this property (namely that white can always play such that the intersection [j W¡ is in S) necessarily contain continuum-many elements (Lemma 23), that countable intersections of sets with this property again have this property (Theorem 2), and that if S has this property, and f(x) has a continuous derivative with f'(x) # 0 everywhere, then the set of a with /(a) e S again has this property (Theorem 1). These facts imply the result stated at