ON BADLY APPROXIMABLE NUMBERS AND CERTAIN GAMES
ON BADLY APPROXIMABLE NUMBERS AND CERTAIN GAMES
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DOI：
10.1090/s0002-9947-1966-0195595-4
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发表时间：
1966
影响因子：
1.3
通讯作者：
W. Schmidt
中科院分区：
文献类型：
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作者：
W. Schmidt
1. 介绍。如果j A - p/q | > c/q2对于某些c > 0和所有有理数pjq，那么数字A被称为严重近似。已知当且仅当无理数a的连分式的偏分母有界时，无理数a是严重近似的[4，定理23]。在最近的一篇论文[7]中，我证明了以下类型的结果：// fuf2，•■•是导数连续且无处消失的微分函数，那么存在连续多个数a，使得所有数fi(a),f2(a)，•••都是严重近似的。设0<a<l/2, 0< ß < 1，并考虑以下两个玩家黑白的博弈。第一个黑选择闭合区间B′。然后，white选择一个封闭区间wxc By，其长度是a乘以Bt的长度，然后black选择一个封闭区间B2 <= Wx，该区间的长度是IF的长度的ß倍。然后white再选择一个封闭区间w2cb2，长度为a乘以B2，以此类推。如果区间W}的交集非常接近，则称白棋为胜局；否则黑棋为赢家。谁会赢？由于Lebesgue测度为0，[4，定理29]，人们可能会认为黑棋总是能赢。然而事实证明，白棋总是赢（定理3）。我们将证明具有这个性质的集合S（即白总能使S中的交集[j W ' ']）必然包含连续多元素（引理23），具有这个性质的集合的可数交集又具有这个性质（定理2），如果S具有这个性质，并且f(x)处处具有f'(x) # 0的连续导数，那么具有/(a) e S的集合a又具有这个性质（定理1）。这些事实暗示了所述的结果
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