46 problems

For an approximation function $\psi:\mathbb{N}\to[0,\infty)$, consider the set of $\psi$-approximable points in dimension one with inhomogeneous parameter $\gamma=0$, with the addi…

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 $\psi$-…

Allen–Ramírez conjecture. In the inhomogeneous cases with $nm=2$, the monotonicity condition can be removed: if

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…

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

Allen's two-dimensional inhomogeneous Khintchine conjecture. If

Let $b,t\geq 2$ be integers with the same prime divisors, let $D\subseteq\{0,\ldots,b-1\}$ be a digit set containing at least one of $0$ and $b-1$, let $C(b,D)$ be the set of real…

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…

Hussain–Simmons divergence conjecture. If this series diverges, then

Hussain–Simmons conjecture. The Lebesgue measure of $\mathcal M^\times(n,m;\psi)$ depends on the convergence or divergence of this series.

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 $m,n\in\mathbb N$. 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…

Beresnevich–Velani dimension conjecture. The Hausdorff dimension of $\mathbf{Bad}_d(\psi)$ is equal to the Hausdorff dimension of $\mathcal W_d(\psi)$.

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 $\\|qx-\mathbf p\\|<\psi(q)$…

Let $n>1$, let $\boldsymbol\alpha\in\mathbb R^n$, and let $\boldsymbol T=(T_1,\dotsc,T_n)\in[1,+\infty)^n$. Write $\bar T=T_1\dotsm T_n$, and let…

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…

Assume the Khintchine–Groshev theorem holds without monotonicity when $n=2$. Monotonicity-free Hausdorff-measure Khintchine–Groshev conjecture. The final monotonicity condition can…

Let $\boldsymbol{x}$ range over the simultaneously approximable points in the unit cube, with $obreak{}$... Higher-dimensional Duffin–Schaeffer conjecture. For $k dots$

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…