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Statement

Setting (pp. 6--7), in dimension $d=1$. $\alpha$ is irrational. For a bounded measurable $S\subset\mathbb R$, $\chi_S(x)=\sum_{k\in\mathbb Z}\mathbb 1_S(x+k)$ is the multiplicity of its projection to $\mathbb T=\mathbb R/\mathbb Z$, 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 $x\in\mathbb T$ (display (2.1), p. 6).

Theorem 2.6 (p. 9, attributed to Hecke and Ostrowski, quoted). "Any interval $I\subset\mathbb R$ with length in $\mathbb Z\alpha+\mathbb Z$ is a BRS."

The interval is arbitrary in position, and its length may exceed $1$, in which case $\chi_I$ counts multiplicity. Together with Proposition 2.4 this is the one-dimensional Hecke-Ostrowski-Kesten characterization that the paper recalls on pp. 1--2: an interval $I\subset\mathbb R$ is a BRS if and only if its length belongs to $\mathbb Z\alpha+\mathbb Z$.

Read depth. Claims checked: the statement and the definitions it uses were read clause by clause on the page images, and the short proof on p. 9 was read. 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. 9. The bounded remainder property does not depend on the interval's position, so one may take $I=[0,\beta)$ with $\beta\in\mathbb Z\alpha+\mathbb Z$, $\beta\notin\mathbb Z$ (an integer length is trivial). By Proposition 2.5 (p. 8: a BRS for a rotation vector $\beta\in\mathbb Z\alpha+\mathbb Z^d$, $\beta\notin\mathbb Z^d$, is a BRS for $\alpha$) it suffices to treat the rotation by $\beta$, and there $g(x)=-\{x\}$ is a bounded transfer function: $g(x)-g(x-\beta)$ jumps by $+1$ at $0$ and by $-1$ at $\beta$, is constant in between and has integral zero, so it equals $\chi_I-\mathrm{mes}\,I$.

Bears on

• Problem 998: the theorem is the converse of the problem's corrected statement (an interval of length $\{j\alpha\}$ has bounded discrepancy), which the problem page credits to Hecke and Ostrowski. Because the position is arbitrary, it also gives bounded discrepancy for translates whose endpoints are not fractional parts of multiples of $\alpha$. As printed, the theorem bounds the discrepancy for almost every starting point $x$, while the problem counts the orbit from one fixed point.