Uniformly spread discrete sets in \(\mathbb{R}^ d\).

*(English)*Zbl 0774.11038Let \(H\subset\mathbb{R}^ d\). 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.

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.

Reviewer: P.Hellekalek (Salzburg)