32 problems

Univariate inhomogeneous Duffin–Schaeffer conjecture for systems of linear forms. For almost every $\mathbf{x}\in \operatorname{Mat}_{n\times m}(\mathbb{R})$, there exist infinitel…

Shell-wise p-adic Duffin–Schaeffer conjecture. One should have

Let $n\in\mathbb N$, let $\omega=(\boldsymbol{\omega}_j)_{j\geq 1}$ be a sequence in $[0,1]^n$, and let $\Psi=(\psi_1,\ldots,\psi_n)$ be an $n$-tuple of non-increasing functions. W…

Khintchine's moving-target conjecture. If $\psi:\mathbb{N}\tomathbb{R}_{\geq 0}$ is decreasing and $\sum_{q=1}^{\infty}\psi(q)=\infty$, then

Weak Duffin–Schaeffer conjecture with a moving target. For every such $\rho$ and $\mathbf y$,

Weak inhomogeneous Duffin–Schaeffer conjecture. The condition denoted by … , so the mathematical content and current status cannot be checked from the excerpt.

Fix $m,n\in\mathbb N$ and an inhomogeneous parameter $\mathbf y\in\mathbb R^m$. For every $\mathbf q\in\mathbb Z^n$, define … Let $W_{n,m}^P(\psi)$ be the corresponding limsup set…

Fix $m,n\in\mathbb N$. Let $\mathbf y\in\mathbb R^m$ and define, for $\mathbf q\in\mathbb Z^n$, … Here $W_{n,m}^P(\psi)$ is the corresponding limsup set and $\operatorname{m}$ deno…

Let $k\in\mathbb{N}$, let $f$ be a dimension function such that $r^{-k}f(r)$ is monotonic, and let $\mathcal{A}_k”(\psi)\subset[0,1]^k$ be the set of simultaneously reduced $\psi$-…

Let $k\in\mathbb{N}$ and let $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$. Define $\mathcal{A}_k”(\psi)\subset[0,1]^k$ to be the set of points $x$ for which $\\|qx-\mathbf p\\|<\psi(q)$…

Velani's conjecture. If

GCP(1) conjecture. If

One-dimensional Diophantine condition conjecture. Suppose that

Non-Liouville inhomogeneous multiplicative approximation conjecture. If

Let $\Psi:\mathbb{N}\to\mathbb{R}_{\geqslant 0}$ and define $W(\Psi)$ as the set of numbers in $[0,1]$ approximable by $a/n$ within $\Psi(n)$ for infinitely many $n\in\mathbb{N}$ a…

Let $f:\mathbb{N}\to\mathbb{N}_0$ satisfy $0\leqslant f(n)\leqslant n$, let $\Omega_f$ be the associated random space of numerator sets, and let $W^P(\Psi)$ denote the set of numbe…

For $m,n$ positive integers, let $\operatorname{Sing}_{m,n}^*(\omega)$ denote the set of $m\times n$ matrices with Diophantine exponent at least $\omega$ that are not trivially sin…

Let $f$ be an approximation function, let $\theta$ be an inhomogeneous shift, and let $\phi$ and $d$ denote Euler's totient and divisor-counting functions, respectively. Let…

Let $B_n\subseteq\{0,1,\dots,n-1\}$ for each integer $n$. For an approximation function $f$ and an inhomogeneous shift $\theta$, define $W_B(f,\theta)$ as the set of $x\in[0,1]$ sa…

Let $B_n\subseteq\{0,1,\dots,n-1\}$ for each integer $n$. For an approximation function $f$ and an inhomogeneous shift $\theta$, define … Write $\mathbf{0}$ for the zero shift. The…

Let $\Phi:\mathbb N\to\mathbb R_{\geqslant 0}$ satisfy the hypothesis denoted by in the source. Duffin--Schaeffer conjecture. For almost all ${\beta}\in\mathbb R$, the inequality ……

Fiber Khintchine conjecture. If $\psi\colon\mathbb{N}\to\mathbb{R}^+$ is monotonic and $\alpha\in\mathbb{R}$, then

Algebraic partial-quotient conjecture. For every such $\alpha$,

Folklore conjecture. The only algebraic irrationals that belong to $\mathbf{Bad}$ are the quadratic irrationals.

Let $(X,T,\mu)$ be a smooth ergodic measure-preserving dynamical system of dimension $d$. Let $\lambda_1<\ldots<\lambda_k$ be its Lyapunov exponents, and let $d_1,\ldots,d_k$ be th…