Lluís Bibiloni, Pelegrí Viader, Jaume Paradís, "On the concept of optimality interval", International Journal of Mathematics and Mathematical Sciences, vol. 30, Article ID 562705, 9 pages, 2002. https://doi.org/10.1155/S0161171202011420
On the concept of optimality interval
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 , 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 , what is the set of all numbers for which 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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