Fractal dimensions of graph of Weierstrass-type function and local H\"older exponent spectra

We study several fractal properties of the Weierstrass-type function $$W(x)=\sum_{n=0} ^\infty \lambda (x) \lambda(\tau x) \cdots \lambda (\tau ^{n-1}x)\, g(\tau ^n x),$$ where $\tau :[0,1)\to[0,1)$ is a cookie cutter map with possibly fractal repeller, and $\lambda$ and $g$ are functions with proper regularity. In the first part, we determine the box dimension of the graph of $W$ and Hausdorff dimension of its randomised version. In the second part, the Housdorff spectrum of the local H\"older exponent is characterised in terms of thermodynamic formalisms. Furthermore, in the randomised case, a novel formula for the lifted Hausdorff spectrum on the graph is provided.