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Uniformly spread discrete sets in $$\mathbb{R}^ d$$. (English) Zbl 0774.11038
Let $$H\subset\mathbb{R}^ d$$. In the following, we shall denote the cardinality of the set $$H$$ by $$| H|$$, the $$s$$-dimensional Hausdorff measure by $$\lambda_ s(H)$$ and the perimeter by $$p(H)$$, $$p(H)=\lambda_{d-1}(\partial H)$$. A subset $$S$$ of $$\mathbb{R}^ d$$ will be called discrete if every bounded subset of $$S$$ is finite. A discrete set $$S$$ will be called uniformly spread with density $$\alpha$$, if there exists a positive constant $$C$$ such that $\big|| S\cap H|- \alpha\lambda_ d(H)\big|\leq C\cdot p_ 1(H)$ for all bounded measurable subsets $$H$$ of $$\mathbb{R}^ d$$. Here $$p_ 1(H)$$ is the quantity $p_ 1(H):=\lambda_ d \{x: \text{ dist}(x,\partial H)\leq 1\}.$ In this paper, the author generalizes a result of himself on discrete sets in $$\mathbb{R}^ 2$$ [see Theorem 3.1 in J. Reine Angew. Math. 404, 77-117 (1990; Zbl 0748.51017)] to $$\mathbb{R}^ d$$, $$d\geq 2$$. He obtains:
Theorem. For every discrete set $$S$$ in $$\mathbb{R}^ d$$ and for every $$\alpha>0$$ the following statements are equivalent:
(i) $$\exists C>0$$, constant: $$|| S\cap H|-\alpha\lambda_ d(H)|\leq C\cdot p_ 1(H)$$ for all bounded measurable subsets $$H$$ of $$\mathbb{R}^ d$$,
(ii) $$\exists C>0$$, constant: $$|| S\cap H|-\alpha\lambda_ d(H)|\leq C\cdot p(H)$$ for all $$H$$ that are finite unions of unit cubes,
(iii) $$\exists\varphi: S\to \alpha^{-1/d}\mathbb{Z}^ d$$, $$\varphi$$ a bijection, such that $$\sup_{x\in S} |\varphi(x)-x|<\infty$$.
It is open if, in conditions (i) or (ii) above, $$H$$ may be restricted to sets homeomorphic to a ball. In the case $$d=2$$ this is true. Let us call the quantity $$|| S\cap H|-\alpha\lambda_ d(H)|$$ the $$\alpha$$-discrepancy of $$S$$ with respect to $$H$$. In a second theorem, the author shows that, if the $$\alpha$$-discrepancy is small for every lattice cube, then $$S$$ is uniformly spread with density $$\alpha$$. Next, let $$H$$ be a finite union of unit cubes. Further results of this paper allow to estimate the number of lattice cubes that are necessary to construct $$H$$ by the operations of disjoint unions and proper differences.

##### MSC:
 11K38 Irregularities of distribution, discrepancy 51M16 Inequalities and extremum problems in real or complex geometry
##### Keywords:
uniformly spread sets; discrepancy; lattice cubes
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