88 problems

For each non-empty finite set $\mathcal{I}\subset\mathbb{N}$, let … where $a_n(x)$ are the continued-fraction digits of the irrational number $x$. Texan conjecture. The set … is de…

Janicki and Woyczynski's conjecture. For every $t>0$, one has

Let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree, let $G$ be a closed subgroup, and let Hausdorff dimension be computed in $\Gamma(p)$. A subgroup is…

Let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree, and let the closure of a subgroup generated by random elements be considered in the profinite topolo…

Let $\Gamma(p)$ be the profinite automorphism group of the rooted $p$-ary tree, and let positive-dimensional mean having positive Hausdorff dimension as a closed subgroup of…

Hausdorff-dimension conjecture for $\Theta$. The set $\Theta$ in Theorem $$ has Hausdorff dimension at least $2/3$.

Oberlin's conjecture. Then

Conjecture 2.8. If

Curved Kakeya conjecture. If $\phi$ satisfies Bourgain's condition, then every $\phi$-Kakeya set has Hausdorff dimension $n$.

Multiplicative-noise extension conjecture. Theorem (ii) and (iii) for the additive-noise equation remain valid in the presence of multiplicative noise, provided that

Uniform dimension conjecture. The uniform dimension theorem stated for $p\geqslant4$ is valid whenever $p\geqslant2$, provided that

Let $P_t:\mathbb{R}^n\rightarrow\mathbb{R}^m$ be an analytic family of linear maps. Define $f:[0,n]\rightarrow\mathbb{R}$ as follows: for an integer $x\in[0,n]$, let $f(x)$ be the…

Let $P_t:\mathbb{R}^n\rightarrow\mathbb{R}^m$ be an analytic family of linear maps, meaning that its coordinate functions are analytic functions of $t$. Assume that for every subsp…

Falconer's distance set conjecture. If $\dim_{\mathcal{H}}E>\frac{d}{2}$, then

Let $\mathbf{M}_{n,1}$ be the set of nondegenerate curves in $\mathbb{R}^n$; that is, curves $\mathcal{C}=\{\mathbf{f}(x): x\in I,\ \mathbf{f}:I\to \mathbb{R}^n\}$ such that they a…

For $k\in\mathbb{N}\setminus\{1\}$, let $L_{\geq k}$ be the Lüroth set whose digits are all at least $k$. For subsets of $\mathbb{R}$, write $A+B=\{a+b:a\in A,b\in B\}$. The interv…

Let $L(\psi)$ be the set of points defined by the Lüroth approximation condition associated with a function $\psi:\mathbb{N}\to(0,1]$, and let $\underline{\tau}_\psi$ denote the co…

Let $L(\psi)$ be the set of points defined by the Lüroth approximation condition associated with a function $\psi:\mathbb{N}\to(0,1]$, and let $\underline{\tau}_\psi$ denote the co…

Let $E\subseteq\mathbb{R}^2$ be a compact set with Hausdorff dimension … Assume there is a set $R\subseteq\operatorname{SL}_2(\mathbb{R})$ of matrices such that $\theta(E)=E$ for e…

Exact dimension formula. One has

Generalized Kakeya conjecture.

Banaji–Fraser conjecture. Under these assumptions, $\dim$ is the Hausdorff dimension. This proposes that adding Hölder stability to the standard properties of a dimension character…

Let $1\leq d<n$ be integers. Let $\widetilde{\mathcal M}_{n,d}$ be the class of submanifolds $\mathcal M\subset\mathbb R^n$ of dimension $d$ that are nondegenerate at every point,…

Stochastic codimension conjecture. For every Borel set $F$ in the state space,

Let $t$ denote the polynomial used in the $t$-adic Littlewood conjecture, and let $f$ be a growth function. Consider the set of counterexamples to $t$-LC with growth function $f$.…