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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.
Statistics, Econometrics and Quantitative Methods
asymptotic distribution functions
minkowski's function
singular functions
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