88 problems

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

Let $X,Y\subset [0,1)$ be closed and invariant under $T_a,T_b$ respectively, where $T_m(x)=mx\bmod 1$ is multiplication by $m$ on the circle. Assume that $\log a/\log b$ is irratio…

Let $p,q\ge 2$ be integers with $p\nsim q$, where $a\nsim b$ means that $\log a/\log b\notin\mathbb Q$. Let $T_m:[0,1]\rightarrow[0,1]$ be the map $x\mapsto mx\mod 1$, and let…

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…

Oberlin's conjecture. Then

Let $E\subset \mathbb{R}^d$ be a Borel set, let $\dim_H(E)>d-1$, and let $s\leq d-1$ be a real number. Radial projection exceptional-set conjecture. … This conjecture strengthens t…

Sharpness conjecture. For every $\beta<\frac{1}{3}$, there exists a non-conservative weak solution

Let $p$ and $q$ be two multiplicatively independent natural numbers, and let $x\in[0,1)$ be a real number. Define the forward orbit … where $T_q$ is the map on…

Dimension-one conjecture. Under these assumptions,

Let $\mathcal{M}$ be the Mandelbrot set, and write … where $J_c$ is the Julia set of the quadratic map with parameter $c$. Star-shapedness conjecture. The set…

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 $2/3$.

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…

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…