8 problems

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

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 $\\|qx-\mathbf p\\|<\psi(q)$…

Going-up spectrum realization conjecture. There is $\underline{\xi}\in\mathbb{R}^{\mathbb{N}}$ such that, for every $N\geq 1$, its first $N$ coordinates $\underline{\xi}_N$ satisfy

Let $u,v\in\mathbb{R}$ be real numbers, and let $\left< u \right>$ denote the distance from $u$ to the nearest integer. Badziahin–Velani conjecture. For all real numbers $u$ and…

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

Three-dimensional Fibonacci-operator conjecture. Every best approximation other than $\mathcal A_3$ belongs to the set

Let $B$ be the non-hyperbolic operator displayed in the source, and let $(1,0,0)$ be the initial integer vector. The best approximations are considered through the points…