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Statement

For a sequence $(x_k)_{k\ge1}$ on the circle $S^1\cong[0,1)$, the first $n$ terms cut the circle into intervals; an $r$-span is the total length of $r$ consecutive intervals.

Theorem 2 (p. 2). Some sequence $(x_k)_{k\ge1}$ on $S^1\cong[0,1]$, together with one universal constant $c$ with $0<c<\infty$, has this property: for each $r\in\mathbb N$ there is a threshold, depending on $r$, such that for every $n\in\mathbb N$ beyond it the intervals cut by $x_1,\dots,x_n$ satisfy

$$\frac{\text{largest $r$-span}}{\text{smallest $r$-span}} \ \le\ 1+\frac{c\log r}{r}.$$

The paper proves it for two sequences, the golden-ratio Kronecker sequence $x_k=\{k\varphi\}$ with $\varphi=(1+\sqrt5)/2$ and the base-2 van der Corput sequence $\frac12,\frac14,\frac34,\frac18,\ldots$, and says the statement was conjectured in Brethouwer's 2024 Ph.D. thesis.

Source. F. Clément and S. Steinerberger, Balanced stick breaking, arXiv:2511.14637v1 (18 November 2025), Theorem 2 on p. 2 and Theorem 3 on p. 2 of that arXiv version, read in the text layer. The edition read is identified in the source digest.

Read depth. Claims checked: the statement was read clause by clause, with Theorem 3 and the remark that it implies Theorem 2; the proofs (Sections 2--3, pp. 3--11) were not read. Unrefereed; not independently reviewed.

Proof pointer

Theorem 3 (p. 2): for either sequence there is a universal $c>0$ such that for all $r\in\mathbb N$, all $n$ sufficiently large depending on $r$, and all $0\le x\le1-r/n$,

$$\Bigl|\#\bigl\{1\le k\le n:\ x\le x_k\le x+\tfrac rn\bigr\}-r\Bigr|\le c\log r,$$

and the paper says the bound holds for all intervals of length $r/n$ on $S^1$ (p. 3). A short-interval discrepancy bound of this kind controls every $r$-span from both sides, which gives Theorem 2 (the derivation is on p. 9). Section 2 (pp. 3--9) proves Theorem 3 for the van der Corput sequence through an ordering lemma (Lemma 1, p. 4) and Lemmas 2--4 (p. 5); Section 3 (pp. 9--11) proves it for the golden-ratio sequence through the three-distance structure of the Kronecker sequence (Lemma 5, p. 9). Not reconstructed here. The derivation of Theorem 2 from Theorem 3, with the quantifiers made explicit and the literal $r=1$ case of both statements recorded as false, is reconstructed (author-recorded, unreviewed) at its reconstruction page; the proofs of Theorem 3 remain unreconstructed.

Dependencies

Classical properties of the van der Corput and golden-ratio Kronecker sequences; the paper's own short-interval discrepancy estimates.

Bears on

• Problem 1221: with $\mu_r=\inf_a\limsup_nM_n^r(a)/m_n^r(a)$ this gives $\mu_r\le1+c\log r/r$ for every $r\ge2$ (an authored one-line remark; at $r=1$ the printed statement would give $\mu_1\le1$, while $\mu_1=2$), so $r(\mu_r-1)\le c\log r$: the third expression of the de Bruijn--Erdős conjecture grows at most logarithmically if it grows at all. The 1949 lower bound is (5.7), $\mu_r\ge1+1/r$.