191 problems

Let $W_{\beta,\lambda}(x)=\sum_{n=1}^{\infty}\lambda^{-\beta n}e^{2\pi i\lambda^n x}$, where $0<\beta\leq 1$ and $\lambda>1$. For a real-valued function $f$, write…

Mandelbrot's conjecture.

Fässler–Orponen's conjecture. For $\mathcal{H}^1$-almost every $\theta\in[0,1]$, both

Oberlin's conjecture. Then

A collection of line segments in the plane can be replaced by the associated full lines without increasing its Hausdorff dimension. More generally, let a collection of line segment…

Let $K_{1/3}$ be the middle-third Cantor set, let $\tau>1$, and let $W_1(\psi_\tau,0)$ denote the set of points satisfying the corresponding one-dimensional Diophantine approximati…

Let $E \subset \mathbb{R}^2$ be an $\alpha$-Furstenberg set, meaning that for each direction $e \in S^1$, there exists a line segment $\ell$ in direction $e$ such that…

Let $D\subseteq\mathbb{N}$ be an infinite set, and define … Let … be the exponent of convergence of the associated series. Hirst's conjecture. The Hausdorff dimension satisfies … H…

Full Hausdorff dimension conjecture. The set $\boldsymbol{FS}_m$ has full Hausdorff dimension, namely

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…

Minimum-dimension conjecture. The infimum is attained at $\beta_3$.

Let $\VV_n=(x,x^2,\ldots,x^n)$ be the Veronese curve, and consider the Hausdorff dimension of its Diophantine approximation sets. Baker and Schmidt obtained lower and upper bounds…

Palis–Yoccoz conjecture. The Hausdorff dimensions should satisfy

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…

Astala's conjecture. The optimal upper bound for the Hausdorff dimension of a $K$-quasicircle is

Let $G$ be a pro-$p$ group, and let $\operatorname{hspec}^{\mathcal G}(G)$ denote its Hausdorff spectrum with respect to a filtration $\mathcal G$. A pro-$p$ group is $p$-adic anal…

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 $\\{\alpha_n\\}_{n\in\mathbb{Z}_+}$ be Verblunsky coefficients satisfying … for some $\gamma\in(0,1)$, and let the associated Szegő matrices be evaluated at…

Let $0\leq k<n$ be integers and let $s\in[0,n]$. A compact set $B\subset\mathbb{R}^n$ is said to contain a $k$-skeleton of an axis-parallel $n$-cube centered at every point of a co…

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

A Schottky group is called nonclassical if it is not a classical Schottky group, and its limit set has a Hausdorff dimension. Lower-bound conjecture. The Hausdorff dimensions of no…

Conjecture B. The set ${\cal E}({\mathbb Z}_3)$ has Hausdorff dimension zero.

Slice double-point conjecture. If $d\in\{4,5\}$, then, outside a single null set,

Exactness conjecture. The lower bound