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On the concept of optimality interval
Viader, Pelegrí; Paradís, Jaume; Bibiloni, Lluís
Universitat Pompeu Fabra. Departament d'Economia i Empresa
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 inboth cases these `Optimality Sets' are intervals and we give aprecise description of their endpoints.
Statistics, Econometrics and Quantitative Methods
diofantine approximations
continued fractions
metric theory
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