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Games and Measures

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Part of the Lecture Notes in Mathematics book series (LNM, volume 785)

Abstract

Suppose that 0 < α < 1, 0 < β < 1. Consider the following game by players Black and White. First, Black picks a compact real interval B1 of length ℓ(B1). Next, White picks a compact interval W1 ⊂ B1 of length ℓ(W1) = αℓ(B1). Then Black picks a compact interval B2 ⊂ W1 of length ℓ(B2) = βℓ(W1), etc. In this way, a nested sequence of compact intervals
$$ B_1 \supset W_1 \supset B_2 \supset W_2 \supset \ldots $$
is generated, with lengths
$$ \ell (B_k ) = (\alpha \beta )^{k - 1} \ell (B_1 ) and \ell (W_k ) = (\alpha \beta )^{k - 1} \alpha \ell (B_1 ) (k = 1,2,3, \ldots ). $$
It is clear that \( \mathop \cap \limits_{k = 1}^\infty B_k = \mathop \cap \limits_{k = 1}^\infty W_k \) consists of a single point.

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References

  1. J.W.S. Cassels (1956). On a result of Marshall Hall. Mathematika 3 109–110.zbMATHMathSciNetCrossRefGoogle Scholar
  2. W.M. Schmidt (1966). On badly approximable numbers and certain games. Trans. A.M.S. 123, 178–199.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1980

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