Paper #226

Title:

A new light on Minkowski's? $(x)$ function

Authors:

Pelegrí Viader, Jaume Paradís and Lluís Bibiloni

Date:

June 1997

Abstract:

The well--known Minkowski's? $(x)$ function is presented as the asymptotic distribution function of an enumeration of the rationals in (0,1] based on their continued fraction representation. Besides, the singularity of ?$(x)$ is clearly proved in two ways: by exhibiting a set of measure one in which ?ï$(x)$ = 0; and again by actually finding a set of measure one which is mapped onto a set of measure zero and viceversa. These sets are described by means of metrical properties of different systems for real number representation.

Keywords:

Asymptotic distribution functions, Minkowski's function, singular functions

JEL codes:

C00

Area of Research:

Statistics, Econometrics and Quantitative Methods

Published in:

Journal of Number Theory, 73, (1998), pp. 212-227

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