Paper #466

Title:

On the concept of optimality interval

Authors:

Pelegrí Viader, Jaume Paradís and Lluís Bibiloni

Date:

May 2000

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.

Keywords:

Diofantine approximations, continued fractions, metric theory

JEL codes:

C00

Area of Research:

Statistics, Econometrics and Quantitative Methods

Published in:

International Journal of Mathematics and Mathematical Sciences, 30, 9, (2002), pp. 559-567

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