Hausdorff dimension of images and graphs of some random complex series

Abstract

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 increasing sequence with $\sup_nλ_{n+1}/λ_n<\infty$ and $φ_n$ satisfies some uniform Lipschitz and boundedness conditions. This class of series includes the famous Weierstrass and Riemann functions as well as others appeared in literature. These results help predict the exact values of the deterministic cases.

Figures (6)

• Figure 1: Curves $W_{\beta, \lambda}([0,1])$ for different $\beta$ and $\lambda$
• Figure 2: Curves $R_{a,b}([0, 1])$ for different $a$ and $b$.
• Figure 3: Curves $W^{\phi}_{\beta,\lambda,\Theta}([0, 1])$ for $\phi(x)=e^{2\pi ix}-\lambda^{-\beta} e^{2\pi i\lambda x}$ and an equidistributed sequence $\theta_n=n\pi\pmod 1$.
• Figure 4: Sample paths of $W^{\phi}_{\beta,\lambda,\Theta}([0, 1])$ for $\phi(x)=e^{2\pi ix}-\lambda^{-\beta} e^{2\pi i\lambda x}$ and $\theta_n\overset{\mathrm{iid}}{\sim} U[0,1]$.
• Figure 5: Curves $W_{\beta,\lambda}^{\phi}([0, 1])$ for $\phi(t)=\lVert t\rVert+i\sin(2\pi t)$.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 2.1: Kolmogorov-Chentsov theorem
• Theorem 2.2: Marcinkiewicz-Zygmund inequality
• Theorem 2.3
• proof
• Proposition 3.1
• Lemma 3.2