4 problems

Let $\mathcal T$ be a tiling set, and call $\Lambda_{\mathcal T}$ a universal spectrum if every set $\Omega$ that tiles $\mathbb R^d$ by $\mathcal T$ is spectral with spectrum…

Let $L$ be the system size, and let $d_{+}^{\mathrm{generic}}$ denote the generic value of $d_{+}$. An aperiodic set $J$ modulo $L$ is a set whose orbit under translation modulo…

Let $\Omega\subseteq\mathbb{R}^{n}$ be a periodic set and let $B\subseteq\mathbb{R}^{n}$ be a periodic half space. Periodic Gaussian surface-area conjecture. The Gaussian surface a…

Periodic tiling conjecture. If a region $\Omega\subset\mathbb{R}^n$ tiles $\mathbb{R}^n$ by translations, then it has a periodic tiling.