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

Jaume Paradís (), Pelegrí Viader () and Lluís Bibiloni
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Pelegrí Viader: https://www.upf.edu/web/econ/faculty/-/asset_publisher/6aWmmXf28uXT/persona/id/3418630

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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