Working Paper 378

A singular function and its relation with the number systems involved in its definition
Jaume Paradís, Pelegrí Viader and Lluís Bibiloni
April 1999
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.
Singular function, number systems, metric number theory
JEL codes:
Area of Research:
Statistics, Econometrics and Quantitative Methods
Published in:
Journal of Mathematical Analysis and Applications, 253, (2001), pp.107-125
With the title:
The Derivative of Minkowski's Singular Function

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