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A singular function and its relation with the number systems involved in its definition;
The Derivative of Minkowski's Singular Function
Paradís, Jaume; Viader, Pelegrí; Bibiloni, Lluís
Universitat Pompeu Fabra. Departament d'Economia i Empresa
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.
15-09-2005
Statistics, Econometrics and Quantitative Methods
singular function
number systems
metric number theory
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