187 problems

Cone-and-cylinder Fourier-dimension conjecture. Then

For the classical Takagi function $T:[0,1]\to\mathbb{R}$, $T(x)=\sum_{n=0}^{\infty}2^{-n}\operatorname{dist}(2^n x,\mathbb{Z})$, and the classical Weierstrass functions…

Let $T:[0,1]\to\mathbb{R}$ be the classical Takagi function, $T(x)=\sum_{k=0}^{\infty}2^{-k}\operatorname{dist}(2^k x,\mathbb{Z})$. For $y\in[0,\tfrac{2}{3}]$, define…

Let $d\ge 1$ and let $\Phi=\{\phi_i\}_{i=1}^m$ be a finite iterated function system of similitudes on $\mathbb{R}^d$ with non-singleton attractor $K$. Suppose that $\Phi$ satisfies…

Let $\mu_{\gamma,\mathrm{GMC}}$ be the one-dimensional Gaussian multiplicative chaos measure on the circle associated with a log-correlated Gaussian field. The Garban--Vargas conje…

For $0<\lambda<\tfrac12$, let $K_\lambda$ be the attractor of the iterated function system $\{x\mapsto\lambda x,\ x\mapsto\lambda x+1-\lambda\}$, let…

Yu's conjectures assert, for the relevant pairs of Cantor sets and their radial projections or division sets, that the corresponding sets have nonempty interior. The supplied sourc…

Let $g$ be a nonconstant real-valued trigonometric polynomial, and let $S(a;q)$ denote the associated square-class chirp functional at the rational point $a/q$, as defined in the W…

Let $X=(X_t)_{t\ge 0}$ and $Y=(Y_t)_{t\ge 0}$ be independent Markov processes taking values in a common metric measure space $(E,d,\mu)$. Define the collision-time and collision-po…

Let $K=K(n,m,\Gamma)\subset\mathbb{R}^{2}$ be a Bedford–McMullen carpet with $n>m\ge 2$ and $1<\#\Gamma<nm$. For every isometry $f:\mathbb{R}^{2}\to\mathbb{R}^{2}$, if…

Let $\{\varphi_i(x)=r_i x+t_i\}_{i=1}^m$ be a finite self-similar iterated-function system on $\mathbb{R}$, where $0<|r_i|<1$, and let $(p_i)_{i=1}^m$ be positive weights with…

Bugeaud–Durand conjecture.

Let $E\subset\mathbb{R}^d$ and let $\mathcal{V}\subset G(d,d-1)$ be Borel sets such that $\dim_{\mathcal{H}}\mathcal{V}>0$ and $\mathcal{V}^{\perp}$ is not contained in a great cir…

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 $\mu$ be the self-similar measure on $\mathbb{R}$ associated to an IFS $\{\varphi_i:i\in\Lambda\}$ without exact overlaps and a probability vector $\{p_i\}$. Let $\lambda_i$ be…

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…

Levesley–Salp–Velani conjecture.

Let $0<u\leq1$ and $0\leq v\leq3$. A sine wave $(u,v)$-Furstenberg set or circular $(u,v)$-Furstenberg set is a set $F\subseteq\mathbb{R}^2$ of the type specified in the paper, wit…

Let $k\in\mathbb{N}$, and let $\mathcal{K}$ be a proper missing-digit fractal in $[0,1]^k$ that is not contained in any proper affine subspace. Let $\mu$ be its Cantor–Lebesgue mea…

Let $\gamma:[0,1]\to\mathbb S^2$ be a $C^2$ curve satisfying … For each $\theta$, let $p_{\gamma(\theta)}$ denote projection onto the line in direction $\gamma(\theta)$, and let…

Let $\Phi = \{x\mapsto r_i x + a_i\}_{i\in {\mathcal I}}$ be a self-similar iterated function system on $\mathbb R$, with $|r_i|\in (0,1)$, and let $\Lambda_\Phi$ be its attractor.…

Assouad dimension conjecture. If

Fractal-dimension conjecture. The above notion of fractal dimension for self-similar subsets of $O_K$ is well-defined and well-behaved with respect to inclusion: fractal dimension…

Hensley's conjecture. The denominators contain every sufficiently large natural number if and only if the Hausdorff dimension exceeds one half:

Fix $n,m\in\mathbb{N}$ and let … for $A,B\subseteq\mathbb{N}_0$. Let $r,s\in\mathbb{N}$ be multiplicatively independent, and let $A,B\subseteq\mathbb{N}_0$ be $\times r$- and…