9 problems

Let $W_{\beta,\lambda}(x)=\sum_{n=1}^{\infty}\lambda^{-\beta n}e^{2\pi i\lambda^n x}$, where $0<\beta\leq 1$ and $\lambda>1$. For a real-valued function $f$, write…

Mandelbrot's conjecture.

Let … For $A\subset\mathbb{R}$, let $G_{W_{\beta,\lambda}}(A)=\{(x,W_{\beta,\lambda}(x)):x\in A\}$. Dimension conjecture. For all $\lambda>1$ and $0<\beta\leq 1$, for every Borel s…

For a function $g$ and parameters $\alpha$ and $b$, let $W_g^{\alpha,b}$ denote a prevalent $\alpha$-Weierstrass function on $\mathbb{S}^1$, and let $\operatorname{dim_H}(A)$ denot…

For $b\in\mathbb{N}$, $0<\alpha<1$, and a $\mathbb{Z}$-periodic Lipschitz function $g$, let … and denote the corresponding family of Weierstrass-type functions by…

Let $W_{a,b}:cmathbb{R}\to\mathbb{R}$ be the Weierstrass function … where $b\in\mathbb{N}$ and $1/b<a<1$. Let $D$ denote the Hausdorff dimension of the graph of $W_{a,b}$. Weierstr…

Let $\rho$ be the SBR measure and let its explicit density formula be the one given in the cited proposition. The paper proves square integrability of the density under its transve…

Let $\rho$ be the SBR measure associated with the Weierstrass system, and let the transversality methods refer to the methods used in the paper to establish smoothness and boundedn…

Upward-jump representation conjecture. The representation formula by means of upward jump times of dyadic expansions has the potential to shed more light on the complex arguments u…