38 problems

Let $A\subseteq \mathrm{SO}(3,\mathbb{R})$ be open, and let $\mu$ denote normalized Haar measure on $\mathrm{SO}(3,\mathbb{R})$. For a unit vector $u\in\mathbb{R}^3$, define the an…

Exact value conjecture. The universal constant $\delta_{\mathcal K}$ is the only real root of

Zero-measure conjecture.

Measure-sequence characterization. The sequences of measures defining a density in this sense are precisely those satisfying

Let $\mathbf L$ be a Lebesgue graph, meaning a graph on the standard probability space $[0,1]$ whose edge set is Lebesgue-measurable. A finite graph $F$ has a density in a graph gi…

Let $\psi$ be a dyadic orthonormal wavelet, let $f:\mathbb{R}\to\mathbb{R}$ tile $obreakmathbb{R}$ by translations and dilations, and let $mu$ denote Lebesgue measure. Define a me…

Let $S_n$ be a syntactic fractal space with a family of definable sets $cF_n$, and let $\nu_n$ and $\mu_n$ be $S_n$-measures. Write $\nu_n \ll \mu_n$ when $\mu_n(A)=0$ implies…

Let $p>0$, and for a probability measure $\mu$ on $\mathbb{S}^{d-1}$ define the $p$-frame energy by … A minimizer is a probability measure attaining the minimum of this energy. Bil…

Let $\mu$ be the infinite-dimensional Lebesgue measure on $\mathbb{R}^{\mathbb{N}}$, and let $\mathfrak{c}$ denote the cardinality of the continuum. A $\mathfrak{c}$-separable enve…

Let $\bar{\eta}_1$ and $\bar{\eta}_2$ be cumulative-effort constraints satisfying $0<\bar{\eta}_1<\bar{\eta}_2$, and let $\mu_{\alpha_{opt},\bar{\eta}_i}$ be the measures associate…

A set $X\subset\mathbb{R}$ is full measure universal if every set of full Lebesgue measure contains an affine copy $\lambda X+t$ with $\lambda\in\mathbb{R}\setminus\{0\}$ and…

For $X\subseteq\mathbb{R}$, call $X$ strongly meager if $X+M\neq\mathbb{R}$ for every Lebesgue measure zero set $M\subseteq\mathbb{R}$. Full measure universality characterization c…

A subset $X\subseteq\mathbb{R}$ is full measure universal if every Lebesgue measurable subset of $\mathbb{R}$ whose complement has Lebesgue measure zero contains an affine copy of…

Let $X\subseteq\mathbb{R}$ be a measure universal set, meaning that every Lebesgue measurable subset of $\mathbb{R}$ with positive Lebesgue measure contains an affine copy of $X$.…

Probabilistic representer conjecture. There exists an $\mathbb{F}$-measure $\nu$ on $X$, with support contained in the support of $\mu$, such that

Let $\mathrm{Col}:\mathbb N\to\mathbb N$ be the Collatz map, and let $\mu$ be a finitely additive $\mathrm{Col}$-invariant measure on $\mathbb N$, with every singleton measurable.…

Convex refractor existence conjecture. Under these hypotheses, such a convex weak solution exists for the target set and data described above.

Let $\mu$ and $\lambda$ be Borel measures on $S^{n-1}$, with $\lambda$ absolutely continuous. Say that $\mu$ is weak Aleksandrov related to $\lambda$ when it satisfies the weak Ale…

Fix a network and a tail-exponent vector ${\boldsymbol{\gamma}}$. Let ${\boldsymbol{\lambda}}^*$ range over nonnegative nominal arrival vectors, and let 0-RJF and G-RDS denote the…

Let $a,b\in \mathbb{R}$ with $0<a<b$. Let $(n_k)_{k \in \mathbb{N}}$ be a strictly increasing sequence of positive integers such that … A non-trivial signed measure with prescribed…

Fix the combinatorics $G$ of a periodic $2$-colored tiling of the plane, and consider the parameters of locally foldable tilings with those fixed combinatorics. Measure-zero conjec…

Weak converse of Zeckendorf's theorem. Given a finite list of positive integers … -Zeckendorf subset of $$ that is not an open interval has measure zero.

Let $cmu$ be a measure on $cmathbf{R}^2$ that charges no line, and suppose that its support has Hausdorff dimension strictly greater than $1$. Let $cell_cmu$ be the order-type limi…

Fejes Tóth's continuous energy conjecture. The energy integral is maximized by $\nu_{ONB}$:

Let $\mu$ be a measure under consideration, and let $\mathcal{S}_{nt}^{sing}(\mu)$ denote its non-tangential singular set. Meagreness conjecture. The set…