12 problems

Let $\mathcal{W}^{\alpha,b}$ be the Banach space of $1$-periodic functions considered above. For a function $W_{\mathbf{t}}$ in this space, let … be the measure on its graph. For…

Simon's conjecture. The absolutely continuous spectrum of $H_V$ is the positive real axis:

Let $d=3$, and let $\Gamma$ be a periodic curve for which a decomposition into translates of a suitable period cell $\mathcal{P}$ satisfying hypothesis (c) does not exist, for exam…

Let a partially hyperbolic system have a center foliation $W^c$, when it exists. Typical-system conjecture. For typical partially hyperbolic systems, the center foliation is not ab…

Consider a discrete-time triangular Pólya urn satisfying conditions A0–A2, A-5, and A-6–A-7, but not necessarily A-8. The relevant limits may be zero with positive probability, so…

Varjú's conjecture. For $\lambda>1/2$, the condition that $\lambda^{-1}$ is not a Pisot number should characterize the absolute continuity of $\mu_\lambda$.

Let $\sigma$ be a nonempty compact subset of $\mathbb{C}$, let $f\in AC(\sigma)$, and let $\varepsilon>0$. Write $Z(f)=\{z\in\sigma:f(z)=0\}$. Small-variation conjecture. There is…

Ngai–Wang conjecture. The measure $\nu_{\lambda_1,\lambda_2}^{1/2}$ is absolutely continuous for almost all $\lambda_1,\lambda_2\in(0,1)$ such that $\lambda_1\lambda_2>1/4$.

For $<\lambda<1$, let $\nu_\lambda$ denote the corresponding Bernoulli convolution, namely the distribution of the random series … where the $\epsilon_n$ are independent random sig…

Fix a permutation $p$ and write $\sigma(p,w)(e^{i\theta})=e^{i\Sigma(\theta)}$, where $\Sigma$ is the associated nondecreasing lift. Let…

Let $\mu$ be a self-similar measure on $\mathbb{R}$, and let $\operatorname{s-dim}\mu$ denote its similarity dimension. Assume that … and that the defining iterated function system…

Let $\nu_{\beta}$ be the Bernoulli convolution associated with $\beta$, and let $f_n$ denote the sequence of approximating functions introduced in the paper. Convergence conjecture…