Sergii Favorov
Uniformly spread discrete sets
We consider discrete infinite sets in $\mathbb R^p$, i.e. infinite sets without finite limit points. Such sets appear in various branches of analysis (zero and pole sets of almost elliptic functions, various models in the mathematical theory of quasicrystals, and so on). In my talk I am going to introduce some notions and to present some theorems connected with such sets.
For arbitrary infinite discrete sets $A=(a_n)_{n\in I}$ and $B=(b_n)_{n\in I}$ put $d(A,B)=\inf\sup_n|a_n-b_{\sigma(n)}|$, where $\inf$ is taken over all bijections
$\sigma$ of the index set $I$. Actually, the condition $d(A,B)<\infty$ means that $B$ is a bounded perturbation of the set $A$.
Theorem 1. If the function $f(t)=d(A,A+t)$ is bounded uniformly in $t\in\mathbb R^p$, then $A$ has a uniform density $\Delta=\Delta(A)$, $0<\Delta<\infty$, i.e.
$$\exists\lim_{T\to\infty}\frac{\text{card}\{n:\,\|a_n-c\|<T\}}{(2T)^p}=\Delta,$$
uniformly with respect to $c\in\mathbb R^p$, where $\|\cdot\|$ is the $l^\infty$-norm in $\mathbb R^p$. Moreover, $d(A,\Delta^{-1/p}\mathbb Z^p)<\infty$.
Corollary 1. Let $L_1,L_2$ be arbitrary full-rank lattices in $\mathbb R^p$. Then $L_1$ is a bounded perturbation of $L_2$ iff $L_1$ and $L_2$ have the
same density.
Next, a discrete set $A$ is Bohr's almost periodic, if its counting measure $\sum_{x\in A}\delta_x$ is Bohr's almost periodic in the weak sense. We find a geometric criterium for any discrete set to be Bohr's almost periodic. Also, we prove the following result.
Theorem 2. If a discrete set $A\subset\mathbb R^p$ is almost periodic and the set of differences $A-A$ is discrete, then $A=L+F$, where $F$ is finite, and $L$ is a full-rank lattice in $\mathbb R^p$.
A similar result was obtained for sets whose counting measure is Besicovitch's almost periodic in the weak sense.