4 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 $\psi$-…

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

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