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

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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.

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  • Pelegrí Viader & Jaume Paradís & Lluís Bibiloni, 1997. "A new light on Minkowski's? $(x)$ function," Economics Working Papers 226, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:226
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    Keywords

    Asymptotic distribution functions; Minkowski's function; singular functions;
    All these keywords.

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General

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