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Statement

Setting (pp. 2--3, 6--7). $\alpha=(\alpha_1,\dots,\alpha_d)\in\mathbb R^d$ is an irrational vector: $1,\alpha_1,\dots,\alpha_d$ are linearly independent over the rationals. A bounded measurable $S\subset\mathbb R^d$ is a bounded remainder set (BRS) if some constant $C=C(S,\alpha)$ satisfies $\bigl|\sum_{k=0}^{n-1}\chi_S(x+k\alpha)-n\,\mathrm{mes}\,S\bigr|\le C$ for $n=1,2,3,\dots$ and almost every $x\in\mathbb T^d$, where $\chi_S(x)=\sum_{k\in\mathbb Z^d}\mathbb 1_S(x+k)$ (display (2.1), p. 6); it is Riemann measurable if its boundary has measure zero (p. 7).

Theorem 4 (p. 5, quoted). "For a convex polytope $S$ in $\mathbb R^d$ to be a bounded remainder set, it is necessary that $S$ is centrally symmetric and has centrally symmetric $(d-1)$-dimensional faces."

Corollary 4 (p. 5). A convex polyhedron $S$ in $\mathbb R^3$ with vertices in $\mathbb Z\alpha+\mathbb Z^3$ is a BRS if and only if it is a zonohedron, that is, centrally symmetric with centrally symmetric faces. The paper obtains it (p. 30) from Theorem 4 and Corollary 1 (p. 2, recorded on the Theorem 1 page), since in dimension three the two classes of polytopes coincide; it notes that for $d\ge4$ the class in Theorem 4 is strictly larger than the zonotopes.

Read depth. Claims checked: Theorem 4, Corollary 4 and the proof on p. 30 were read clause by clause on the page images; the cited result of Mürner and Theorem 5.1 were not checked. Nothing here is independently reviewed.

Source. Sigrid Grepstad and Nir Lev, Sets of bounded discrepancy for multi-dimensional irrational rotation, Geom. Funct. Anal. 25 (2015), no. 1, 87--133, doi:10.1007/s00039-015-0313-z, read in arXiv:1404.0165v2 as identified on the source card; pages are those of the arXiv version.

Proof pointer

P. 30. By Theorem 5.1 (p. 27) a bounded remainder polytope has vanishing Hadwiger-type invariants for the group $\mathbb Z\alpha+\mathbb Z^d$; these force the classical Hadwiger invariants (for all translations) to vanish, and a result of Mürner, which the paper cites, says that for a convex polytope this is equivalent to central symmetry of the polytope and of its $(d-1)$-dimensional faces.

Bears on

No Erdős problem page in the corpus.