A New Light on Minkowski's ?(x) Function

Pelegrí Viader; Jaume Paradís; Lluís Bibiloni

doi:10.1006/jnth.1998.2294

A New Light on Minkowski's ?(x) Function

Pelegrí Viader, Jaume Paradís, Lluís Bibiloni

Department of Teaching of Mathematics and Experimental Sciences

Research output: Contribution to journal › Article › Research › peer-review

48 Citations (Scopus)

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. The singularity of ?(x) is proved in two ways: by exhibiting a set of measure one in which ?′(x)=0; and again by actually finding a set of measure one which is mapped onto a set of measure zero and vice versa. These sets are described by means of metrical properties of different systems for real number representation. © 1998 Academic Press.

Original language	English
Pages (from-to)	212-227
Journal	Journal of Number Theory
Volume	73
Issue number	2
DOIs	https://doi.org/10.1006/jnth.1998.2294
Publication status	Published - 1 Dec 1998

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AB - 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. The singularity of ?(x) is proved in two ways: by exhibiting a set of measure one in which ?′(x)=0; and again by actually finding a set of measure one which is mapped onto a set of measure zero and vice versa. These sets are described by means of metrical properties of different systems for real number representation. © 1998 Academic Press.

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JO - Journal of Number Theory

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A New Light on Minkowski's ?(x) Function

Abstract

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