4 problems

Let $\mathcal{R}_{s,\beta,\mu}$ be the convex set of Riesz point processes. Suppose that it is not reduced to one point, and let $\mathscr{P}_j$ be a finite collection of simple po…

Let $\mathcal{X}_{s,\mu}$ denote the Riesz equilibrium infinite configurations. For $s>d$, let $r=\mu^{-1/s}$; for $s<d$, let $r=\rho_b^{-1/d}$, where $\rho_b>0$ is the background…

Let $s\in(-2,\infty)\setminus\\{d\\}$, and let $e(s)$ denote the minimal energy per unit volume of the Riesz gas. For a Bravais lattice $\mathscr{L}$ with unit-volume fundamental c…

Number-rigidity conjecture. Number-rigidity should occur for $s\leq d-1$ at any dimension $d$.