41 problems

Dimension formula conjecture. Under these assumptions,

Łaba–Wang conjecture. If the self-similar measure $\mu_{b,D}$ is a spectral measure, then $\alpha D$ is an integer tile for some $\alpha\in\mathbb{R}$.

Let $R\in M_n(\mathbb{Z})$ be an expanding matrix, and let $D,L\subset\mathbb{Z}^n$ be finite digit sets with $\#D=\#L$. The pair $(R,D)$ is admissible when $(R,D,L)$ is a Hadamard…

Let $\mathcal{F}$, $\mathcal{K}$, and $\mu$ satisfy the hypotheses of Theorem 1, and suppose that the fractal $\mathcal{K}$ contains a dense set of rational vectors. Let the upper…

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…

Let … is a spectral measure, meaning that $L^2$ has an orthogonal basis of complex exponentials. Let $=_j$ be the digit set. The spectral self-similar measure conjecture. Then…

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…

Let $1<\beta<2$ be a Pisot number other than the golden ratio, and let $\mu_{\mathbf{p}}$ be the self-similar measure considered in the paper. Non-doubling conjecture. The measure…

Let $\nu$ be a self-similar measure supported on $[1,\infty)$, and consider the sequence $\{x^n\bmod 1\}_{n\in\mathbb N}$ for $x$ sampled from $\nu$. Baker's conjecture. The sequen…

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

Exact overlaps conjecture. In the self-similar setting,

Exact overlaps conjecture. If $\Phi$ has no exact overlaps, then

Let … mu$. Consider the sequence$({x^n}){ngeq 1}$modulo one. **Baker's conjecture.** For every such self-similar measure, the sequence$({x^n}){ngeq 1}…

Modified Łaba–Wang conjecture. If $\mu_{\rho,\mathcal{D},{\bf p}}$ is a spectral measure, then ${\bf p}$ is equal-weight, with $p_j=1/m$; $\rho=1/N$ for some integer $N\geq2$; and…

Let $N\ge 2$ be a positive integer, let $\mathcal{D}$ be a digit set defining the self-similar measure $mu_{N,\mathcal{D}}$, and let a product-form Hadamard triple with respect to…

Beurling-density upper-bound conjecture. For any maximal orthogonal set $\Lambda$ of $\mu_{p,q}$,

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…

Tribonacci parameter conjecture. The minimizer is the tribonacci parameter:

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…

Exact-overlap conjecture. For iterated function systems acting on $\mathbb{R}$, the only mechanism preventing equality in the similarity dimension inequality, the similarity dimens…

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…

Minimal loop-class conjecture. For every real $q$,