44 problems

Let $k\in\mathbb{N}$, let $f$ be a dimension function such that $r^{-k}f(r)$ is monotonic, and let $\mathcal{A}_k”(\psi)\subset[0,1]^k$ be the set of simultaneously reduced $\psi$-…

Finite Hausdorff measure conjecture. If

Integer-power conjecture. The only dimension functions $\alpha$ for which there exists an $\alpha$-set $E$ supporting a measure $\mu$ satisfying this Fourier decay condition are in…

Exact Hausdorff-measure conjecture. There exist positive finite constants $C_1$ and $C_2$ such that

Let $M$ be a smooth manifold of dimension $3$ equipped with a bracket-generating distribution $\Delta$, and for $x\in M$ let $\mathcal{X}^x_{\Delta}$ be the set of endpoints of sin…

Let $d\in(0,2]$, let $E\subset\mathbb C$ be compact, and let $h$ be a principal $K$-quasiconformal mapping conformal on $\mathbb C\setminus E$. Here $\mathcal M^d$ denotes $d$-dime…

Let $E\subset \mathbb C$ be compact, let $0<t\leq 2$, and let $h$ be normalized and conformal in $\mathbb C\setminus E$, admitting a $K$-quasiconformal extension to $\mathbb C$. He…

Negative-slice conjecture. Outside a single null set, the slice need not be a finite set for every $s>0$; indeed, there may exist a non-trivial measure function $\phi$ such that

Hussain–Simmons divergence conjecture. If this series diverges, then

Equality conjecture. If $A$ is a strictly self-similar fractal with dimension $d$, then

Besicovitch's conjecture. If $\Theta^1_*(E,x)>\frac{1}{2}$ for $\mathcal{H}^1$-almost every $x\in E$, then $E$ is countably $1$-rectifiable.

Positive critical Hausdorff measure conjecture. With probability one with respect to $\mu_{\mathbf{p}}$,

Let $X\in\text{Alex}^n(-1)$. For $0\le k\le n-1$, define the $k$-singularity function by setting $\theta_k(x)=2\pi$ for $x\in\mathcal{S}^{k-1}(X)$ and…

Let $\text{Alex}^n(-1)$ denote the $n$-dimensional Alexandrov spaces with curvature at least $-1$. For $0\le k\le n-1$, let $\mathcal{S}^k_\epsilon(X)$ be the set of points whose t…

Numerical conjecture. The spherical Hausdorff measure satisfies

Let $E$ be a subset of a metric space, let $\mathscr{H}^\alpha$ denote the $e3e3alpha$-dimensional Hausdorff measure, and let $\mathscr{H}^\alpha\llcorner E$ be its restriction to…

Invariance conjecture. For every compact Polish space $X$ with

Hausdorff-measure lower-bound conjecture. There exists a constant $C>0$, depending on $n$, $k$, $hR$, and an upper bound for $\mathscr{H}^k(E)$, such that

Manifold Khintchine-type conjecture. The Hausdorff $g$-measure of $\dualappr_{n,\bm f}^*(\psi_0,\ldots,\psi_m)$ is zero if $S_{n,d}^g(\psi_0,\ldots,\psi_m)$ converges, and equals…

Let $n\in\mathds{N}$, let $k$ be a nonnegative integer, let $(X,p)\in\operatorname{Alex}^n(-1)$, and set … Here $\mathcal{S}^k_{\epsilon_i}$ denotes the $\epsilon_i$-singular strat…

Let $n\in\mathds{N}$, let $k$ be a nonnegative integer, and let $(X,p)\in\operatorname{Alex}^n(-1)$, where $\mathcal{S}^k_\epsilon$ denotes the $\epsilon$-singular stratum. Hausdor…

Skewed projection inequality. If $A$ is a weak antichain, then

Let $n\geq 1$. An antichain in $[0,1]^n$ is a subset whose distinct points are incomparable in the coordinatewise order, and let $\mathcal H^{n-1}$ denote $(n-1)$-dimensional Hausd…

Let $\boldsymbol{D}_0$ be embedded in $\boldsymbol{D}$ under the setting of Proposition, and let $\mathcal M^{\alpha}$ denote the Hausdorff measure used in that setting. Embedded-s…

Let $\boldsymbol{D}$ be the random metric object in the unimodular Frostman setting, let $\mathcal H^{\alpha}_{M}(\boldsymbol{D})$ denote its unimodular Hausdorff measure, and let…