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Statement

Setting (pp. 2, 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. For a bounded measurable $S\subset\mathbb R^d$, $\chi_S(x)=\sum_{k\in\mathbb Z^d}\mathbb 1_S(x+k)$ is the multiplicity of its projection to $\mathbb T^d=\mathbb R^d/\mathbb Z^d$, and $S$ 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 $x\in\mathbb T^d$ (display (2.1), p. 6). The parallelepiped spanned by linearly independent $v_1,\dots,v_d\in\mathbb R^d$ is $P=\{\sum_{k=1}^d t_kv_k:0\le t_k<1\}$ (p. 2); it need not project injectively to the torus.

Theorem 1 (p. 2, quoted). "Any parallelepiped in $\mathbb R^d$ spanned by vectors $v_1,\dots,v_d$ belonging to $\mathbb Z\alpha+\mathbb Z^d$ is a bounded remainder set."

Theorem 3.1 (p. 11) restates it with the addition that $P$ admits a Riemann integrable transfer function, a bounded $g$ on $\mathbb T^d$ with $\chi_P(x)-\mathrm{mes}\,P=g(x)-g(x-\alpha)$ almost everywhere.

Consequences stated in the paper.

• Corollary 1 (p. 2; proof p. 18). Every convex, centrally symmetric polygon in $\mathbb R^2$ with vertices in $\mathbb Z\alpha+\mathbb Z^2$, and more generally every zonotope in $\mathbb R^d$ with vertices in $\mathbb Z\alpha+\mathbb Z^d$, is a BRS.
• Corollary 2 (p. 3; proof p. 18, as Proposition 3.7). For every positive $\gamma=n_0+n_1\alpha_1+\cdots+n_d\alpha_d$ with integers $n_j$ there is a bounded remainder parallelepiped spanned by vectors in $\mathbb Z\alpha+\mathbb Z^d$ with measure $\gamma$; if moreover $\gamma\le1$ it may be chosen simple (projecting injectively to $\mathbb T^d$). With Proposition 2.4 this shows the positive measures of bounded remainder sets are exactly the positive numbers of that form.
• Theorem 3.8 (p. 19; proof pp. 24--25) widens the class: with $v_1,\dots,v_d\in\mathbb Z\alpha+\mathbb Z^d$, the parallelepiped spanned by $w_1=v_1$ and $w_k\in v_k+\mathrm{span}\{v_1,\dots,v_{k-1}\}$ ($2\le k\le d$) is a BRS.

Read depth. Claims checked: the statement, Theorem 3.1, Corollaries 1 and 2 and Theorem 3.8 were read clause by clause on the page images. The proof was located but 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

§§3.1--3.6, pp. 11--17 (for $d\ge2$; $d=1$ is Theorem 2.6). The transfer function is built explicitly. Its Fourier coefficients are forced by the cohomological equation (§3.1); the formal gradient of that series is identified (Lemma 3.3, p. 14) with a surface measure on an oriented, piecewise-linear closed hypersurface $\Pi$ in $\mathbb T^d$ made of $(d-1)$-dimensional parallelepipeds determined by $P$, minus a constant vector times Lebesgue measure. The function $g$ is then defined as an intersection number with $\Pi$ minus a linear term (display (3.21), p. 16); it is piecewise linear with jumps on $\Pi$, hence Riemann integrable, and comparing Fourier series ends the proof (§3.6, p. 17).

Bears on

• Problem 998: in dimension one the theorem says only that an interval with an endpoint at $0$ and length in $\mathbb Z\alpha+\mathbb Z$ is a BRS, a case of the Hecke-Ostrowski sufficiency that the paper states for every interval as Theorem 2.6, the converse of the problem's corrected statement, which the problem page credits to Hecke and Ostrowski. The paper presents Theorem 1 as the higher-dimensional extension of that result; its content for $d\ge2$ does not bear on the problem.