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On the concept of optimality interval

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Abstract

The approximants to regular continued fractions constitute `best approximations' to the numbers they converge to in two ways known as of the first and the second kind. This property of continued fractions provides a solution to Gosper's problem of the batting average: if the batting average of a baseball player is 0.334, what is the minimum number of times he has been at bat? In this paper, we tackle somehow the inverse question: given a rational number P/Q, what is the set of all numbers for which P/Q is a `best approximation' of one or the other kind? We prove that in both cases these `Optimality Sets' are intervals and we give a precise description of their endpoints.

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  • Pelegrí Viader & Jaume Paradís & Lluís Bibiloni, 2000. "On the concept of optimality interval," Economics Working Papers 466, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:466
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    Keywords

    Diofantine approximations; continued fractions; metric theory;
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    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General

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