Constructing Singular Functions via Farey Fractions

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 x 1 f /stf x., f /s1y 1yt.f x. for all xgw0, 1x, xq1 xq1 F. where f:w0,1xªR is the unknown, and x 1 r /str x., r /stq 1yt.r x. for all xgw0, 1x, R. xq1 2yx where r:w0,1xªR is the unknown. In both cases, tg 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