International Journal of Mathematics and Mathematical Sciences
Volume 30 (2002), Issue 9, Pages 559-567

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, Barcelona, Spain
2Departament d'Economia i Empresa, Universidad Pompeu Fabra, Ramon Trias Fargas 25-27, Barcelona 08005, Spain

Received 18 January 2001; Revised 15 June 2001

Copyright © 2002 Lluís Bibiloni et al. 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.


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.