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A new light on Minkowski's? $(x)$ function
Viader, Pelegrí; Paradís, Jaume; Bibiloni, Lluís
Universitat Pompeu Fabra. Departament d'Economia i Empresa
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.
2005-09-15
Statistics, Econometrics and Quantitative Methods
asymptotic distribution functions
minkowski's function
singular functions
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