The Hausdorff dimension of the nondifferentiability set of the Cantor function is [(2)/(3)]²

The main purpose of this note is to verify that the Hausdorff dimension of the set of points N* at which the Cantor function is not differentiable is [ln(2)/ln(3)]2. It is also shown that the image of N* under the Cantor function has Hausdorff dimension ln(2)/ ln(3). Similar results follow for a standard class of Cantor sets of positive measure and their corresponding Cantor functions. The Hausdorff dimension of the set of points N* at which the Cantor function is not differentiable is [ln(2)/ ln(3)]2. Chapter 1 in [5] provides a nice introduction to Hausdorff measure and dimension; references [5-7] pursue the topic. We begin our proof with some notation and discussion. Let C denote the Cantor set. Let N+ (N-) denote the set of points at which the Cantor function does not have a right side (left side) derivative, finite or infinite. Then N* = N+ U NU {t: t is an end point of C} denotes the nondifferentiability set of the Cantor function. Although we will assume familiarity with [4], where Eidswick characterized N*, some material is repeated for completeness. A number t in C has a ternary representation t = (t1, .I . , t1, . . .), where ti = 0 or 2. Let z(n) denote the position of the nth zero in the ternary representation of t; (la) If t E N+, then limsup{z(n + 1)/z(n)} > ln(3)/ln(2); (lb) If lim sup{z(n + 1)/z(n)} > ln(3)/ ln(2), then t E N+. Let md denote the d-dimensional Hausdorff measure, and put r = ln(2)/ ln(3). We will compute the Hausdorff dimension of N* by verifying (A) If 1 > d > r2, then mdN* = 0. (B) If d Kd > 0; Kd will be specified later for a sequence of d 's increasing to r2. Condition (A) will be verified for each d satisfying the inequalities 1 > d > r2 by constructing a set E (depending on d) which contains N* and satisfies the equation mdE = 0. To verify (B), we will consider a sequence {dn } of Received by the editors May 24, 1991 and, in revised form, January 13, 1992. 1991 Mathematics Subject Classification. Primary 26A30, 28A78. ? 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page