12 problems

Let $\gamma=\alpha+\mathrm i\beta\in\mathcal D$, where $\mathcal D$ is the domain of parameters for which the Gaussian multiplicative chaos measures $\mathrm M_{\gamma}$ and…

Inhomogeneous Duffin--Schaeffer conjecture.

Let $W(1,1;\psi,\theta)$ be the inhomogeneous well-approximable set, and suppose the dimension formula … where … holds under the stated hypotheses on $\psi$. Chen--Xiong's conjectu…

Cai--Hambrook's conjecture. If

Let $B^H=(B_t^H)_{t\in[0,1]}$ be a fractional Brownian motion with Hurst parameter $H\in(0,1)$, and let … be its graph. Fraser–Sahlsten conjecture. Almost surely, … Fraser and Sahl…

Let $M_\gamma$ be the standard sub-critical Gaussian multiplicative chaos measure on the unit circle, with parameter $0<\gamma<1/\sqrt{2}$, and let $\dim_F(M_\gamma)$ denote its Fo…

A hypersurface $M$ in Euclidean space has principal curvatures, the eigenvalues of its Weingarten map, at each point. Its Fourier dimension, denoted by $\dim_F(M)$, measures the st…

Let $p>2$ be a prime number and let $\epsilon\in(0,1)$. Uniform Fourier-dimension conjecture. For every digit set $D$ with … we have … where the $o(1)$ term is uniform across all s…

Let $d\geq 1$ and define the cone … For a set $A\subseteq\mathbb{R}^{d+1}$, its Fourier dimension is … where $\mathcal{P}(A)$ is the set of Borel probability measures on…

Zero-measure conjecture for $(d,k)$-sets. No $(d,k)$-sets with $d>k\geqslant 2$ and zero $d$-dimensional Lebesgue measure exist.

Let $d$ be a positive integer, let $RY=\{ry:r\in R,\ y\in Y\}$ for $R\subseteq(0,\infty)$ and $Y\subseteq\mathbb{R}^d$. Dual fractal-set conjecture. For every $\alpha\in[0,1]$, the…

Let $d$ be a positive integer, let $\beta\beta\in[0,d]$, and let $RY=\{ry:r\in R,\ y\in Y\}$ for $R\subseteq(0,\infty)$ and $Y\subseteq\mathbb{R}^d$. Fractal-set conjecture. For ev…