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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). Bounded remainder sets for a second irrational vector $\beta$ are defined the same way with $\beta$ in place of $\alpha$.

Theorem 5 (p. 5). Let $\alpha,\beta$ be irrational vectors in $\mathbb R^d$ and $T$ an invertible linear map of $\mathbb R^d$. The image under $T$ of every Riemann measurable BRS for $\alpha$ is a BRS for $\beta$ if and only if

$$T(\mathbb Z\alpha+\mathbb Z^d)\subset\mathbb Z\beta+\mathbb Z^d .$$

Corollary 5 (p. 5). For irrational vectors $\alpha,\beta$ in $\mathbb R^d$, every BRS for $\alpha$ is a BRS for $\beta$ if and only if $\alpha\in\mathbb Z\beta+\mathbb Z^d$. This one is not restricted to Riemann measurable sets: sufficiency is Proposition 2.5 (p. 8) and necessity is Theorem 5 for the identity map (proof p. 33).

The paper also parametrizes the pairs $(\beta,T)$ satisfying the condition by the $(d+1)\times(d+1)$ integer matrices with nonzero determinant (Theorem 6.1, p. 34), and proves sufficiency for bounded remainder sets that need not be Riemann measurable under the stronger condition $T(\mathbb Z\alpha+\mathbb Z^d)=\mathbb Z\beta+\mathbb Z^d$ (Theorem 6.2, p. 35).

Read depth. Claims checked: Theorem 5, Corollary 5 and the proof on pp. 32--33 were read clause by clause on the page images; Theorems 6.1 and 6.2 are reported from the paper's own summary on pp. 5 and 32 only. 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

§6.1, pp. 32--33. Sufficiency: by Corollary 3 a Riemann measurable BRS for $\alpha$ is equidecomposable to a parallelepiped spanned by vectors of $\mathbb Z\alpha+\mathbb Z^d$ by translations from that group, and $T$ carries the whole picture to the $\beta$ side, where Corollary 3 applies again. Necessity ($d\ge2$; $d=1$ follows from the Hecke-Ostrowski-Kesten characterization): if $Tv\notin\mathbb Z\beta+\mathbb Z^d$ for some $v\in\mathbb Z\alpha+\mathbb Z^d$, the parallelepipeds $P_t$ spanned by $v$ and $v_k+tv$ ($t\in\mathbb R$, with $v,v_2,\dots,v_d$ independent in $\mathbb Z\alpha+\mathbb Z^d$) are bounded remainder sets by Theorem 3.8, so their images are bounded remainder sets for $\beta$; the vertex condition of Theorem 5.4 (p. 30) on those images then puts uncountably many vectors into the countable group $\mathbb Z\beta+\mathbb Z^d$.

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