Working Paper 378

Title:
A singular function and its relation with the number systems involved in its definition
Authors:
Jaume Paradís, Pelegrí Viader and Lluís Bibiloni
Date:
April 1999
Abstract:
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.
Keywords:
Singular function, number systems, metric number theory
JEL codes:
C00
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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