On the Hausdorff dimension of some graphs

Abstract:

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