24 problems

Let $C(b,D)$ be a generalized Cantor set with Hausdorff dimension $cgamma$, with $D$ containing at least one of $0$ and $b-1$. Let $t\geq 2$ be an integer such that the sets of pri…

Let $C(b,D)$ be a generalized Cantor set with Hausdorff dimension $cgamma$, where $b$ and $t$ have the same prime divisors, $D$ contains at least one of $0$ and $b-1$, and…

Pluriharmonic-measure dimension conjecture. The dimension of pluriharmonic measure of domains in $\mathbb{C}^n$ is at most

Let $T\subseteq \mathbb{Y}$ be a subtree, and write … For $d>\frac{1}{2}$ and $\delta\in\left[\frac{1}{2},d\right]$, consider win-lose alternating-move games on $T$ with payoff set…

Let $\mathcal{L}_{1/2}^{\widetilde{\mathbb{Z}}^3}$ be the loop soup on the metric graph of $\mathbb{Z}^3$, and let $\mathcal{L}_{1/2}^{\mathbb{R}^3}$ be the corresponding loop soup…

Let $\Gamma$ be a non-elementary hyperbolic group generated by a finite set $S$, and let $\nu$ be a probability measure on $S$ with $\operatorname{supp}(\nu)=S$. Consider the branc…

Entropy-minimal subshift conjecture. There exists an entropy minimal subshift representing $A$ if and only if, for every arithmetic progression $P$,

Let $S_n$ denote the number of holes created by the earthworm after $n$ steps, and interpret the dimension of the set of holes through the growth exponent of its expected size. Dim…

Let a hyperbolic set have stable and unstable slices, and interpret “fractal dimension” as either Hausdorff dimension or upper box dimension. Hasselblatt–Schmeling conjecture. The…

Let $0\le \theta<1$ and let $E\subset\mathbb{R}^n$. Write $\dim_A^\theta(E)$ for the Assouad spectrum with parameter $\theta$, and let the global quasiconformal Assouad spectrum di…

Given a number $\beta\in(1,2)$, let $\nu_{\beta}$ denote the associated Bernoulli convolution and let $\dim_H$ denote Hausdorff dimension. Large-degree Pisot dimension conjecture.…

Dimension conjecture. For all $0 < \lambda < \mu < 1$ with $\lambda + \mu > 1$, the set $A_{\lambda,\mu}$ has dimension strictly greater than $1$.

Let $\nu_{\beta_1}$ denote the Bernoulli convolution associated with $\beta_1$. Suppose that $\beta_1\in(1,2)$, $\beta_1\neq(1+\sqrt{5})/2$, and that $\beta_1$ has a real conjugate…

Let $d\in\{4,5\}$, and consider the cable-graph clusters in $(0,1)^d\cap\delta\mathbb Z^d$ as $\delta\to0$. Four- and five-dimensional scaling-limit conjecture. The limit in distri…

Let $d=3$, and let the cable-graph loop-soup clusters be formed on finer and finer meshes. Denote by $\Delta$ the Hausdorff dimension of the Brownian loop-soup clusters. Three-dime…

Let $\alpha$ and $\beta$ be real numbers such that $1,\alpha,\beta$ are independent over the field of rational numbers. An $\alpha\beta$-sequence is a sequence $\{x_n\}_{n\geq 1}$…

Let $\alpha$ be an irrational number and $d$ be a real number. Consider the sequence $\{2^n d+n\alpha\mod 1\}_{n\geq 1}$ in $[0,1]$. Closure-interval conjecture. The topological cl…

Hausdorff-measure conjecture. If $\mathit{PH}_0(\operatorname{supp}\mu)$ is trivial and $0<\alpha<d$, then $$

Let $X_n$ be a random sample of $n$ points from a compact set $X\subseteq\mathbb{R}^m$, distributed according to a nonsingular probability measure $\mu$, where $m\geq 2$. Let…

Let $S(X_t)=\overline{\{x:X(t,x)>0\}}$ be the closed support of $X_t$, let $\partial S(X_t)$ denote its boundary, and let $L_t$ be the local-time measure associated with the bounda…

Let $G$ be the relevant group of similarities of $\mathbb{R}^{d}$, and let $\Phi\subseteq G$ be an affinely irreducible iterated function system with attractor…

Let $A(y)$ be the almost periodic filter represented by a polynomial $P$, let $M(A)=M(P)$ be its Mahler measure, and let $\lambda$ be the scaling parameter. For a Borel measure $f$…

Let $d706>0$ and let $d6fc\in(0,1)\setminus\mathbb{Q}$ be an irrational frequency. Denote by $d6a9_{\lambda,\alpha}$ the phase-independent spectrum of the corresponding Sturmian Sc…

Exact-overlaps conjecture. A strict inequality in the preceding box-dimension bound can occur only in the presence of exact overlaps.