International Journal of Mathematics and Mathematical Sciences

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International Journal of Mathematics and Mathematical Sciences / 2002 / Article

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Volume 30 |Article ID 562705 | 9 pages | https://doi.org/10.1155/S0161171202011420

On the concept of optimality interval

Lluís Bibiloni,1 Pelegrí Viader,2 and Jaume Paradís2

1Facultat de Ciències de l'Educació, Universidad Autònoma de Barcelona, 08193 Bellaterra, Spain

2Departament d'Economia i Empresa, Universidad Pompeu Fabra, Ramon Trias Fargas 25-27, Spain

Received18 Jan 2001
Revised15 Jun 2001

Abstract

The approximants to regular continued fractions constitute best approximations to the numbers they converge to in two ways known as 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.

Copyright © 2002 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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