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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). A bounded set is Riemann measurable if its boundary has measure zero (p. 7).

Proposition 2.4 (p. 8). If $S$ is a BRS, then there are integers $n_0,n_1,\dots,n_d$ with

$$\mathrm{mes}\,S=n_0+n_1\alpha_1+\cdots+n_d\alpha_d .$$

The paper says (p. 8) that for an interval on $\mathbb R$ this is Kesten's theorem, and the introduction (p. 3) calls it a generalization of Kesten's theorem. No regularity beyond the standing boundedness and measurability is assumed.

Companion facts used with it (p. 7, the paper's own short proofs, after Petersen). Proposition 2.1: a bounded measurable $S$ whose discrepancy sums are bounded in $n$ for each $x$ in a set of positive measure is a BRS. Proposition 2.2: for a bounded Riemann measurable $S$, boundedness in $n$ at one single point $x$ already makes $S$ a BRS.

Consequence for intervals (an observation of this page, not printed in the paper). Take $d=1$, $\alpha$ irrational, and $0\le u<v\le1$ with $v-u<1$. Suppose $\#\{1\le m\le n:\{\alpha m\}\in[u,v)\}=n(v-u)+O(1)$ for all large $n$. The count is the discrepancy sum of $I=[u,v)$ at the point $x=\alpha$, and the finitely many smaller $n$ change nothing, so the sums are bounded in $n$ at that point. $I$ is Riemann measurable, so $I$ is a BRS by Proposition 2.2, and Proposition 2.4 gives $v-u=n_0+n_1\alpha$. Since $0<v-u<1$, $n_1\ne0$ and $v-u=\{n_1\alpha\}$. This is the necessity half of Kesten's length criterion, the corrected statement of Problem 998.

Read depth. Claims checked: the statement, the definitions it uses and Propositions 2.1 and 2.2 were read clause by clause on the page images. The proof (p. 8) was read: it rests on the cited fact that every eigenvalue of the irrational rotation by $\alpha$ has the form $\exp 2\pi i\langle n,\alpha\rangle$, $n\in\mathbb Z^d$, which the paper does not prove. 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. 8. By Proposition 2.3 (p. 8), a BRS has a bounded measurable transfer function $g$ with $\chi_S(x)-\mathrm{mes}\,S=g(x)-g(x-\alpha)$ almost everywhere. Because $\chi_S$ is integer valued, $\tau=\exp 2\pi i g$ is an eigenfunction of the rotation with eigenvalue $\exp 2\pi i\,\mathrm{mes}\,S$, and the known form of the rotation's eigenvalues gives the integers. The paper credits the argument to Furstenberg, Keynes and Shapiro and to Petersen.

Bears on

• Problem 998: with Proposition 2.2 it gives, as worked out above, the necessity half of the problem's corrected statement (a bounded-discrepancy interval $[u,v)$ with $0\le u<v\le1$ and $v-u<1$ has $v-u=\{j\alpha\}$ for some integer $j$), which the problem page credits to Kesten's Theorem 4. It constrains only the length and says nothing about the endpoints, which the site's wording asks about.