A new description of uniformly spread discrete sets

TL;DR

This work establishes a dimension-free criterion for uniform spreading: a discrete set $A\subset{\mathbb R^d}$ with bounded translation changes is a bounded perturbation of a lattice ${D^{-1/d}{\mathbb Z}^d}$ for some density $D>0$, i.e., $A$ is roughly shift-invariant and uniformly spread with a lattice-model bijection $\Theta:\mathbb N\to D^{-1/d}{\mathbb Z}^d$ and $\sup_{a\in A}|a-\Theta(a)|<C$. It also proves that, uniformly in $x$, $\#(A\cap B(x,R))=D m_d(B(x,R))+O(R^{d-1})$ as $R\to\infty$, and extends these results to discrete multisets, yielding analogous density and uniform-spread conclusions. Among applications, the supports of Fourier quasicrystals with unit masses are shown to be uniformly spread (and almost periodic in the unit-mass case), with a precise lattice-replication relation; a related graph-theoretic corollary about infinite graphs is discussed. The paper closes with open questions about higher-dimensional analogues and the optimal perturbation bound.

Abstract

We prove that each discrete set in the Euclidean space that has bounded changes under every translation is a bounded perturbation of a square lattice, i.e., a uniformly spread set in the sense of Laszkovich. In particular, the support of every Fourier quasicrystal with unit masses is uniformly spread.

Theorems & Definitions (14)

• Definition 1
• Theorem 1
• Theorem 2
• Corollary
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 3
• Definition 2