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 $d\in(0,2]$, 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…

Finite Hausdorff measure conjecture. If

Hausdorff measure conjecture. 1. If $h_{\mathcal{F}}<2\theta_{\mathcal{F}}$, then

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

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

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 $\psi$-…

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…

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 $A_2,A_3\subset[0,1]$ be closed sets invariant under multiplication by $2$ and $3$ modulo $1$, respectively, and suppose … For $g(x)=\exp(-(-\log x)^s)$, Hausdorff-measure conj…

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…

Let $\psi:\mathbb{N}\to\mathbb{R}^+$ be a decreasing approximation function, and define … Here $\|\cdot\|$ denotes distance to the nearest integer. Let $k$ be the codimension of an…

Let $m,d\geq1$, let $\Psi:\mathbb Z^m\to(0,\infty)$ be arbitrary, and let $f$ be a dimension function such that the source's series … diverges and $x\mapsto x^{-md+1}f(x)$ is monot…

Higher-dimensional Hausdorff-measure divergence conjecture. If $\Psi(\mathbf q)=\psi(|\mathbf q|)$ for a monotonically decreasing $\psi:\mathbb N\to(0,\infty)$, this series diverge…

Let $m,d\geq1$ be integers, let $\Psi:\mathbb Z^m\to(0,\infty)$, and fix $\bftheta\in\I^d$. For $X=(\mathbf x^{(1)},\ldots,\mathbf x^{(d)})\in\I^{md}$, define…

Let $\psi:\mathbb N\to[0,\infty)$ be an approximating function, let $\bftheta\in\mathbb R^d$, and let $\WW^d_{\psi,\bftheta}\subset\mathbb R^d$ denote the set of points $\xx$ for w…

Generalized Baker–Schmidt problem for Hausdorff measure.

Let $\psi:\mathbb{N}\to\mathbb{R}$ be non-negative, let $W(\psi)$ be the associated limsup set of reduced rational approximation intervals, and let $f:[0,\infty)\to[0,\infty)$ be a…