20 problems

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

Allen's two-dimensional inhomogeneous Khintchine conjecture. If

Let $k\geqslant 2$, let $\alpha_1,\ldots,\alpha_{k-1},\gamma_1,\ldots,\gamma_k\in\mathbb R$, and set $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_{k-1})$. Assume that its multiplic…

Inhomogeneous Gallagher divergence conjecture. If

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…

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…

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.

Inhomogeneous Littlewood-type counting conjecture. There exists $G=G(\alpha,\gamma,\delta)\subseteq\operatorname{Bad}(\delta)$ with Hausdorff dimension

Ramírez's inhomogeneous Duffin–Schaeffer conjecture. If the displayed series diverges, then for almost all $\alpha\in[0,1]$, the inequality

One-dimensional Diophantine condition conjecture. Suppose that

Non-Liouville inhomogeneous multiplicative approximation conjecture. If

Let $\delta\in\mathbb R$, and let $\Phi:\mathbb N\to\mathbb R_{\geqslant 0}$ satisfy the two conditions denoted by … in the source. Inhomogeneous Duffin–Schaeffer theorem. Then for…

Let $\boldsymbol{\alpha} \in \mathbb R^n$ and let $d \in \{0,\ldots,n-1\}$. For each $\boldsymbol{\theta} \in \mathbb R^{n-d}$, the inhomogeneous exponent…

Laurent–Nogueira's conjecture. There exists $C>0$ such that, for every irrational $\alpha\in\mathbb{R}$ and every $\gamma\in\mathbb{R}$, there are infinitely many coprime integer p…

Let $\boldsymbol{i}=(i_1,\ldots,i_n)$ be a weight vector, let $\mathcal{S}_n(\boldsymbol{i})$ denote the set of $\boldsymbol{i}$-singular vectors, and for each $\varepsilon>0$ defi…

Let $\psi:\mathbb N\to[0,\infty)$ be an approximating function, let $\bftheta\in\mathbb R^d$, and let $\WW^d_{\psi,\bftheta}\subset\mathbb R^d$ denote the set of points $\xx$ for w…

Winning criterion conjecture.

Cassels' logarithmic conjecture. Almost every pair $(x_1,x_2)\in\mathbb{R}^2$ satisfies, for every $(\gamma_1,\gamma_2)\in\mathbb{R}^2$,

Harrap's conjecture. The set ${\rm BAD}^{\Theta}(\alpha,\beta)$ should have full Hausdorff dimension in $\mathbb{R}^2$, without the additional assumption…