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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). Here $d=2$.

Theorem 3 (p. 4). Let $S$ be a convex polygon in $\mathbb R^2$. Then $S$ is a BRS if and only if it is centrally symmetric and every pair of parallel edges $e,e'$ satisfies both of the following:

1. some point of $e$ and some point of $e'$ differ by a vector in $\mathbb Z\alpha+\mathbb Z^2$;
2. if the midpoints of $e$ and $e'$ do not differ by a vector in $\mathbb Z\alpha+\mathbb Z^2$, then the edge vectors $e,e'$ themselves lie in $\mathbb Z\alpha+\mathbb Z^2$.

The paper notes (p. 4) that both conditions hold when the vertices of $S$ lie in $\mathbb Z\alpha+\mathbb Z^2$. Theorem 5.3 (p. 28) restates the theorem as the vanishing of all rank 0 and rank 1 Hadwiger-type invariants of $S$ for the group $\mathbb Z\alpha+\mathbb Z^2$, and p. 28 explains the equivalence.

Read depth. Claims checked: the statement and its restatement as Theorem 5.3 were read clause by clause on the page images. The proof was read but not checked step by step. 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

§5.5, pp. 28--30. Necessity is Theorem 5.1 (p. 27): the Hadwiger-type invariants for the group $\mathbb Z\alpha+\mathbb Z^d$ of a bounded remainder polytope vanish. Sufficiency goes by induction on the number of edge pairs. A parallelogram is handled by Theorem 3.8, and a polygon whose edge vectors all lie in $\mathbb Z\alpha+\mathbb Z^2$ by Corollary 1 (p. 2, recorded on the Theorem 1 page). Otherwise the polygon is cut into five pieces: three are reassembled by translations in $\mathbb Z\alpha+\mathbb Z^2$ into a parallelogram spanned by vectors of that group, a BRS by Theorem 1, and the other two into a convex polygon with one edge pair fewer, a BRS by induction; Proposition 4.1 carries both back.

Bears on

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