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A singular function and its relation with the number systems involved in its definition

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Minkowski's ?(x) function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it permits us to prove that its derivative, as it also happens for many other non-decreasing singular functions from [0,1] to [0,1], when it exists can only attain two values: zero and infinity. It is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than k* =5.31972, and ?'(x) exists then ?'(x)=0. In the same way, if the same average is less than k**=2 log2(F), where F is the golden ratio, then ?'(x)=infinity. Finally some results are presented concerning metric properties of continued fraction and alternated dyadic expansions.

Suggested Citation

  • Jaume Paradís & Pelegrí Viader & Lluís Bibiloni, 1999. "A singular function and its relation with the number systems involved in its definition," Economics Working Papers 378, Department of Economics and Business, Universitat Pompeu Fabra.
  • Handle: RePEc:upf:upfgen:378
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    Keywords

    Singular function; number systems; metric number theory;

    JEL classification:

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

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