Grepstad lev 2014 bounded discrepancy rotation

../

corollary_3: Grepstad and Lev's characterization of the Riemann measurable bounded remainder sets for the rotation by an irrational vector alpha: exactly the sets equidecomposable, by translations in Z alpha + Z^d, to a parallelepiped spanned by vectors of Z alpha + Z^d.

proposition_2_4: Grepstad and Lev's form of Kesten's theorem: the measure of every bounded remainder set for the rotation by an irrational vector alpha is an integer combination of 1, alpha_1, ..., alpha_d; for an interval in dimension one this is Kesten's necessity.

theorem_1: Grepstad and Lev's first main result: for an irrational vector alpha in R^d, every parallelepiped spanned by vectors of Z alpha + Z^d has bounded remainder, the d-dimensional extension of the Hecke-Ostrowski theorem.

theorem_2: Grepstad and Lev's second main result: any two Riemann measurable bounded remainder sets of the same measure can be cut into finitely many Riemann measurable pieces and reassembled into each other by translations by vectors of Z alpha + Z^d only.

theorem_2_6: Grepstad and Lev's statement and short proof of the Hecke-Ostrowski theorem: for irrational alpha, every interval of the real line whose length lies in Z alpha + Z is a bounded remainder set, independently of its position.

theorem_3: Grepstad and Lev's characterization of the convex polygons in R^2 that are bounded remainder sets for the rotation by an irrational vector alpha: central symmetry plus two conditions in Z alpha + Z^2 on each pair of parallel edges.

theorem_4: Grepstad and Lev's necessary condition for a convex polytope in R^d to be a bounded remainder set for the rotation by an irrational vector alpha: it is centrally symmetric and its (d-1)-dimensional faces are centrally symmetric.

theorem_5: Grepstad and Lev's description of the invertible linear maps T of R^d that send every Riemann measurable bounded remainder set for an irrational vector alpha to a bounded remainder set for beta: exactly those with T(Z alpha + Z^d) contained in Z beta + Z^d.

theorem_6: Grepstad and Lev's result that every Riemann measurable bounded remainder set for the rotation by an irrational vector alpha has a Riemann integrable solution g of the cohomological equation.

---

Sigrid Grepstad and Nir Lev, Sets of bounded discrepancy for multi-dimensional irrational rotation, arXiv:1404.0165v2 (2014; published Geom. Funct. Anal. 25 (2015), no. 1, 87-133, DOI 10.1007/s00039-015-0313-z, checked against Crossref on 2026-10-07; 39 pp.).

The multi-dimensional theory of bounded remainder sets. Here $\alpha=(\alpha_1,\dots,\alpha_d)$ has $1,\alpha_1,\dots,\alpha_d$ linearly independent over the rationals, and a measurable set $S$ is a bounded remainder set when the discrepancy $D_n(S,x)=\sum_{k=0}^{n-1}\chi_S(x+k\alpha)-n\,\mathrm{mes}\,S$ is at most a constant $C(S,\alpha)$ in absolute value for every $n$ and almost every $x$ (pp. 1--2, 6); in dimension one the condition says that $\alpha$ is irrational. Theorem 1 (p. 2) shows that a parallelepiped in $\mathbb R^d$ whose spanning vectors all lie in $\mathbb Z\alpha+\mathbb Z^d$ has bounded remainder (extending Hecke-Ostrowski), and Corollary 3 (p. 3), drawn from Theorems 1 and 2 and Corollary 2, characterizes the Riemann measurable bounded remainder sets as those equidecomposable to such a parallelepiped using translations by vectors in $\mathbb Z\alpha+\mathbb Z^d$ only. Relevance: Characterizes the Riemann measurable bounded remainder sets for multi-dimensional irrational rotation, the direct generalization of the one-dimensional Hecke-Ostrowski-Kesten characterization, which the paper recalls (p. 2): an interval is a bounded remainder set exactly when its length lies in $\mathbb Z\alpha+\mathbb Z$. The paper's section 2 states the two halves of that criterion with short proofs: Proposition 2.4 (p. 8), that the measure of every bounded remainder set lies in $\mathbb Z+\mathbb Z\alpha_1+\cdots+\mathbb Z\alpha_d$, and Theorem 2.6 (p. 9, Hecke-Ostrowski), that every interval with length in $\mathbb Z\alpha+\mathbb Z$ is a bounded remainder set. Problem 998's corrected statement asks for the necessity half for an interval $[u,v)$ with $0\le u<v\le1$ and $v-u<1$, counted along the orbit from one point. The criterion constrains only the length, so every translate of a bounded remainder interval is again one, and bounded remainder does not force the endpoints to be fractional parts of multiples of $\alpha$, as the site's wording of the problem asks.

The copy read for this card is arXiv:1404.0165v2 (22 October 2014, 39 pp.). Read status: claims checked for every result linked below, read clause by clause on the page images with the definitions of pp. 1--3 and 6--7; the depth of each proof's reading is recorded on its result page, and no proof was checked step by step. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1404.0165), every other right reserved.

Bears on. #998: in dimension one, Proposition 2.4 together with Proposition 2.2 yields the problem's corrected statement (a bounded-discrepancy interval $[u,v)$ with $0\le u<v\le1$ and $v-u<1$ has $v-u=\{j\alpha\}$), a derivation written out on the Proposition 2.4 page and not printed in the paper; Theorem 2.6 is the converse, which the problem page credits to Hecke and Ostrowski. The problem page credits the corrected statement to Kesten. The paper recalls the one-dimensional criterion and does not treat the problem itself, and neither result says anything about the endpoints that the site's wording asks about.

Results. Theorem 1 (p. 2, with Corollaries 1 and 2 and Theorem 3.8); Theorem 2 (p. 3); Corollary 3 (p. 3); Theorem 3 (p. 4); Theorem 4 (p. 5, with Corollary 4); Theorem 5 (p. 5, with Corollary 5); Theorem 6 (p. 6); Proposition 2.4 (p. 8); Theorem 2.6 (p. 9).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.