6 problems

Roy's spectrum conjecture. The description above gives the entire spectrum ${\mathcal{S}}_{m,n}$ for every integer $m$ with $1\leq m\leq n-1$. The result is already known for $m=1$…

Exact dimension conjecture. For every $\Lambda\in X_n$,

Let ${\boldsymbol{\nu}}=(\alpha_1,\ldots,\alpha_l,-\beta_1,\ldots,-\beta_m)$ encode the weights, and let $(\Lambda,{\boldsymbol{\nu}})$-systems and proper ${\boldsymbol{\nu}}$-syst…

An $(n,0)$-system is the limiting system considered by Schmidt and Summerer in their framework for simultaneous Diophantine approximation. Schmidt–Summerer's template conjecture. S…

Schmidt–Summerer’s regular-systems conjecture. The general optimal lower bound is reached at regular systems.

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…