95 problems

Zero-measure conjecture.

Translation-invariance conjecture. For every $c\in\mathbb{R}$,

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…

Dynkin's conjecture. Every compact set $S\subset {\bf R}^{n}$ carries a non-trivial doubling measure $\mu$.

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

Exact value conjecture. The upper bound for $\delta_{\mathcal H}$ obtained in the paper, which is the relevant solution of a cubic equation and is approximately $0.272$, is in fact…

A measurable set $P\subseteq 2^\omega$ is regular when there are a $G_\delta$ set $Q\supseteq P$ and an $F_\sigma$ set $S\subseteq P$ such that $\mu(S)=\mu(P)=\mu(Q)$. Here a…

Conjecture on convolution measure types. The equality

Erdős's similarity conjecture. Every measure-universal set is finite; equivalently, no infinite set is measure universal. The Lebesgue density theorem shows that every finite set i…

Set-theoretic obstruction conjecture. There could exist a model of ZFC in which $(\mathbb R^{\mathbb N},\mu)$ does not admit a $\mathfrak c$-separable envelope.

Within the weak formulation, let $\nu_\mathbb F{}_m$ be the abstract signed Radon measure replacing the generally non-canonical product $F^\vee_m(\mathbb J)\,e_1(\widehat J_m)$, an…

Let $A\subseteq\mathbb{R}$ be an infinite set. The set $A$ is called an Erdős set if there exists a measurable set $E\subseteq\mathbb{R}$ with positive Lebesgue measure such that ……

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

Bouleau–Hirsch energy image density conjecture. The invertibility of the generalized Malliavin matrix of $f$ should imply that the law of $f$ is absolutely continuous with respect…

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…

Ionascu-type interpolation conjecture. There exists a measurable, nonnegative function $h$ such that $f\leq h\leq\max\{g_1,g_2\}$ a.e. and

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 $P$ be a set in the relevant Euclidean space. A set or pattern $P$ is measure universal if every Lebesgue measurable set $A$ contains an affine copy of $P$. Erdős's similarity…

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…

Let $\{p_n\}_{n=1}^\infty$ be a decreasing sequence with $1<p_n<\infty$ and $p_n\to1$. For each $n$, let $(\hat{k}_n,\hat{\mu}_n)$, $\hat{\boldsymbol{\tau}}_n$, and…

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…

Let $\mathrm{ITM}(r)$ be the full parameter space of interval translation maps with $r$ translation intervals, equipped with its parameter-space measure. An infinite-type map is on…