15 problems

Let $\Phi$ be an iterated function system, let $p$ be a probability vector, and let $\mu$ be the corresponding self-similar measure. The entropy rate is … where $X_1,X_2,\ldots$ ar…

Fourier-decay conjecture. There exists $\sigma>0$ such that

Multiplicative-convolution conjecture. The measure $\mu_1\cdot \mu_2 \dotsb \mu_k$ is absolutely continuous with respect to Lebesgue measure.

Let $G$ be the group acting on $\mathbb{R}^d$ in the self-similar setting, and let $\nu$ be the associated stationary measure of a finitely supported, contracting on average and ir…

Łaba–Wang conjecture. If $\mu_{\rho,\mathcal{D},{\bf p}}$ is a spectral measure, then ${\bf p}$ is equal-weight, so $p_j=1/m$; $\rho=1/N$ for some integer $N$; and there exists…

Let $\mu$ be the self-similar measure on $\mathbb{R}$ associated to an IFS $\{\varphi_i:i\in\Lambda\}$ without exact overlaps and a probability vector $\{p_i\}$. Let $\lambda_i$ be…

Let $\{\varphi_i(x)=\lambda_i x+t_i\}_{i\in \mathcal{I}}$ be an iterated function system of contracting similarities, and let $b\geq 2$ be an integer. Suppose that … for some…

GCP(1) conjecture. If

Let $s$ be the continuous function appearing in the eigenvalue-counting asymptotics … for a Sturm–Liouville problem with a non-constant weight whose primitive is arithmetically sel…

Let $\mu$ be a non-atomic self-similar measure with support contained in $[1,\infty)$. A sequence $(y_n)_{n=1}^{\infty}$ is uniformly distributed modulo one if, for every interval…

Let $\Phi=\{\varphi_j(x)=\lambda_jx+t_j\}_{j=0}^m$ be a self-similar iterated function system on the real line, let $p=(p_j)_{j=0}^m$ be a probability vector, and let $\mu$ be its…

Let $\Phi$ be an iterated function system and let $(p_i)_{i\in I}$ be a probability vector. For a finite word $\mathbf{a}=(a_1,\ldots,a_n)\in I^n$, write…

Let $\mu$ be a self-similar measure with contraction parameters $r_1,r_2$ and weights $m_1,m_2$ satisfying … Write $\lambda_{N,k}$ and $\lambda_{D,k}$ for the Neumann and Dirichlet…

Spectrum equality conjecture. For a self-similar measure $\mu$ and all $t\in[t_{\min},t_{\max}]$,

Multifractal spectrum conjecture. For all $t \in [t_{\min},t_{\max}]$,