On the Hausdorff dimension of some graphs

Abstract

Consider the functions \[ W b ( x ) = ∑ n = − ∞ ∞ b − α n [ Φ ( b n x + θ n ) − Φ ( θ n ) ] , {W_b}(x) = \sum \limits _{n = - \infty }^\infty {{b^{ - \alpha n}}[\Phi ({b^n}x + {\theta _n}) - \Phi ({\theta _n})],} \] where b > 1 b > 1 , 0 > α > 1 0 > \alpha > 1 , each θ n {\theta _n} is an arbitrary number, and Φ \Phi has period one. We show that there is a constant C > 0 C > 0 such that if b b is large enough, then the Hausdorff dimension of the graph of W b {W_b} is bounded below by 2 − α − ( C / ln ⁡ b ) 2 - \alpha - (C/\ln b) . We also show that if a function f f is convex Lipschitz of order α \alpha , then the graph of f f has σ \sigma -finite measure with respect to Hausdorff’s measure in dimension 2 − α 2 - \alpha . The convex Lipschitz functions of order α \alpha include Zygmund’s class Λ α {\Lambda _\alpha } . Our analysis shows that the graph of the classical van der Waerden-Tagaki nowhere differentiable function has σ \sigma -finite measure with respect to h ( t ) = t / ln ⁡ ( 1 / t ) h(t) = t/\ln (1/t) .