A UNIFIED APPROACH FOR NONDIFFERENTIABLE FUNCTIONS

Let psi(x) denote the distance between x and the nearest integer, and fix 0 < a < 1, ab > 1, where b is not necessarily an integer. Then for any sequence theta(n) of phases, the function f(x) = SIGMA(n=0)infini ty a(n)psi(b(n)x + theta(n)) has no right (left) derivative at any poi nt x and 2 + (log a/log b) is the box-counting dimension of the graph off. The crucial step is to obtain the smallest Lipschitz class to whi ch f belongs. (C) 1994 Academic Press, Inc.