8 problems

Let $n$ be a positive integer and let $\alpha_1<\dots<\alpha_n$ be nonzero real numbers. For $x\in\mathbb{R}$, write $\lVert x\rVert_{\mathbb{R}/\mathbb{Z}}$ for the distance from…

Relative spectrum conjecture.

Near-maximal spectrum conjecture.

Let $\mathbf{n}\in\mathbb{Z}^d_{>0}$ be a velocity vector, and let … Here an interior lattice point means a point $\mathbf w\in\operatorname{int}(T_{\mathbf n})\cap\mathbb{Z}^d$, a…

Let $\mathbf{n}\in\mathbb{Z}^d_{>0}$ be a velocity vector, meaning that its entries are distinct and relatively prime, and let … An interior lattice point is a point…

Let $n_0,n_1,\dots,n_d$ be pairwise distinct integers. For each $i$ with $0\leq i\leq d$, consider the distances of $t(n_i-n_j)$ to the nearest integer as $t\in\mathbb{R}$ varies.…

Let $\boldsymbol v=(v_1,\ldots,v_{n-1})$ be a vector of increasing positive integers, and let $\operatorname{gap}(\boldsymbol v)$ and $\lambda_+(\boldsymbol v)$ be the quantities d…

For positive coprime integers $v_1,\ldots,v_d$, let $\operatorname{ML}(v_1,\ldots,v_d)$ denote the maximum distance from an integer that all runners with these velocities can attai…