CONSTRUCTING SINGULAR FUNCTIONS VIA FAREY FRACTIONS

To illustrate some points about continued fractions, H. Minkowski in 1 904 introduced the so-called ?-function. This function and some genera lizations 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 asf(x/x + 1) = tf(x), f(1/x + 1) = 1 (1 - t)f(x) for all x is an element of [0,1], (F) where f:[0,1] --> R is the unknown, and r(x/x + 1) = tr(x), r(1/2 - x) = t + (1 - t)r(x) for all x is an element of [0,1], (R) where r:[0,1] --> R is the unknown. In both cases, t is an element of (0,1) is a given parameter. In the p resent note we give a general construction of singular functions, base d on the Farey fractions and including, as a special case, the Minkows ki function and its generalizations. In contrast to earlier proofs, th e methods presented here do not make explicit use of the theory of con tinued fractions. (C) 1996 Academic Press, Inc.