Hausdorff dimension of images and graphs of some random complex series

Let $\{X_n= e^{2πi θ_n}\}$ be a sequence of Steinhaus random variables, where $θ_n$ are independent and uniformly distributed on $[0,1]$. We compute the almost sure Hausdorff dimension of the images and graphs of the random complex series $S(x)=\sum_{n=1}^{\infty}a_n X_nϕ_n(λ_nx)$, where $λ_n$ is an