The Hausdorff Dimension of the Nondifferentiability Set of a Nonsymmetric Cantor Function

Abstract

Each choice of numbers a and c in the segment (0, (1/2)) produces a Cantor set C ac by recursively removing segments from the interior of the interval [0, 1] so that intervals of relative length a and c remain on the left and right sides of the removed segment, respectively. A Cantor function Φ ac is obtained from C ac in much the same way that the standard Cantor function, Φ, is obtained from the Cantor ternary set. When a = c = (1/3), C ac is the Cantor ternary set, C, and Φ ac is the standard Cantor function, Φ. The derivative of Φ is zero off C, and the upper derivative is infinite on C; the set N = {x E C the lower derivative of Φ is finite} has Hausdorff dimension [ln2/ln3] 2 . In this paper similar results are established for N ac , the nondifferentiability set of Φ ac . The Hausdorff dimension of N ac is the maximum of the real numbers satisfying the following equations: x(ln(1/c)) 2 = ln((α + c)/c) In((a/c) x + 1), and x(ln(1/α)) 2 = ln((α + c)/a) in((c/α) x + 1).