On badly approximable numbers and certain games

Abstract

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 the beginning. We shall discuss games of much greater generality than those mentioned in this introduction. The author is indebted to the referee for very valuable suggestions. 2. (¡5, ®)-games. Let M, Q be sets, Q' a subset of SI, and a a mapping from Q into subsets of M. Call ^-function any function §> which assigns to every Be£l a nonempty set <?>(B) <= Í2 such that for C e &(B), «(C) c a(B).

Sets with Large Intersection Properties

“…Several authors have introduced large classes of sets of Hausdorff dimension s which turn out to have the property that countable intersections of the sets also have dimension s. Such sets are thought of as having a 'large intersection' property in an -dimensional sense. Examples include the 'incompressible sets' of Schmidt [12,13], the 'regular' sets of Baker and Schmidt [1], the '^^-dense' construction of Falconer [7] and constructions using the 'ubiquitous systems' of Dodson, Rynne and Vickers [5]. These constructions are all somewhat technical, but lead to classes with large intersection properties.…”

Diophantine approximation properties of certain infinite sets

“…)-games introduced in [7] and will be fairly simple. The proof of the more difficult Theorem 3 will also involve games and will be given at the end.…”

“…It was shown in [7,8] that badly approximable «-tuples form an a-winning set in R" for 0 < a < {-. (In fact in §6 we will prove a somewhat more precise assertion.)…”

On Irregularities of Distribution

“…Some rather deep results of Schmidt [5,6] can easily be applied to give extremal results about the magnitude of A(n, / ) . THEOREM 6.…”

“…Now choose x u x 2 ,... to be the Van der Corput sequence (see [5], p. 4). It is readily seen that A ( 2 * -l ) = l (k = 1 , 2 , .…”