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

Jaume Paradís (), Pelegrí Viader () and Lluís Bibiloni

Economics Working Papers from Department of Economics and Business, Universitat Pompeu Fabra

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 (search for similar items in EconPapers)
JEL-codes: C00 (search for similar items in EconPapers)
Date: 1999-04
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