Constructing Singular Functions via Farey Fractions

ISSN： 0022-247X

Source： Journal of Mathematical Analysis and Applications, Vol.203, Iss.1, 1996-10, pp. : 127-141

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Abstract

To illustrate some points about continued fractions, H. Minkowski in 1904 introduced the so-called ?-function. This function and some generalizations of it are known to be singular, i.e., strictly monotone with derivative 0 almost everywhere. They can be characterized by systems of functional equations, such as fbiggl({xover x+1}biggr)=tf(x),qquad fbiggl({1over x+1}biggr)=1-(1-t)f(x)qquadhbox{for all $xin[0,1]$},eqno{hbox{(F)}} where f :[0, 1]-< R is the unknown, and rbiggl({xover x+1}biggr)=tr(x),qquad rbiggl({1over 2-x}biggr)=t+(1-t)r(x)qquadhbox{for all $xin[0,1]$.}eqno{hbox{(R)}} where r : [0, 1]-< R is the unknown. In both cases, t∈ (0, 1) is a given parameter. In the present note we give a general construction of singular functions, based on the Farey fractions and including, as a special case, the Minkowski function and its generalizations. In contrast to earlier proofs, the methods presented here do not make explicit use of the theory of continued fractions.