Clément–Steinerberger: Balanced stick breaking

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theorem_2: There is a sequence on the circle and a universal constant c such that for every r ≥ 2 and all large n the largest r-span is at most 1 + c log r/r times the smallest, so μ_r ≤ 1 + c log r/r; proved for the golden-ratio Kronecker and the van der Corput sequences.

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François Clément and Stefan Steinerberger, Balanced stick breaking, arXiv:2511.14637v1 (18 November 2025), math.CO, 12 pages. Unrefereed; no journal reference on arXiv and no published version found (Crossref, 2026-09-27). Suggested key [ClSt25].

Edition read. The copy read for this card is arXiv v1, retrieved from https://arxiv.org/pdf/2511.14637v1; 409,063 bytes. The text layer was read. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2511.14637), every other right reserved.

Read status. Claims checked for Theorem 2 and Theorem 3 (p. 2), read clause by clause; the proofs (Section 2 for the van der Corput sequence, pp. 3--9, and Section 3 for the golden-ratio sequence, pp. 9--11) were not read. Unrefereed; nothing here is independently reviewed.

Overview

The paper reads the first $n$ terms of a sequence $(x_k)$ on the circle as breaks of a circular stick and asks how unequal $r$ consecutive pieces must become. Its Theorem 1 (p. 1) restates the three $r=1$ results of de Bruijn and Erdős ($1/\log2$, $1/\log4$ and the ratio $2$), noting that Ostrowski, Schönhage and Toulmin also established them, and Section 1.2 (p. 2) recalls the general-$r$ lower bound $1+1/r$ of (5.7) and the de Bruijn--Erdős conjecture that $1/r$ can be replaced by $f(r)/r$ with $f(r)\to\infty$, calling the problem "completely open for every $r\ge2$".

Theorem 2 (p. 2) gives the upper bound: for some sequence on $S^1\cong[0,1]$ and one universal constant $c$, each $r\in\mathbb N$ has a threshold, depending on $r$, beyond which the largest sum of $r$ consecutive gaps cut by the first $n$ terms is at most $1+c\log r/r$ times the smallest. Two examples are given, the golden-ratio Kronecker sequence $\{k\varphi\}$ and the base-2 van der Corput sequence; the paper says the theorem was conjectured, with the rate possibly optimal, in Brethouwer's 2024 Ph.D. thesis (not read here). Theorem 3 (p. 2) is the input: for either sequence there is a universal $c$ with $|\#\{k\le n:x\le x_k\le x+r/n\}-r|\le c\log r$ for all $r$, all large $n$ (depending on $r$) and all $x\le1-r/n$, and the remark after it says the bound holds for all intervals of length $r/n$ on $S^1$, which implies Theorem 2 (p. 3, with the derivation on p. 9). Section 1.3 relates this to Schmidt's theorem, which makes $O(\log n)$ the best possible global discrepancy, and to pair correlation. The introduction's stick picture counts $n+1$ pieces after $n$ breaks; Theorem 2 is stated on $S^1$, where $n$ points cut $n$ arcs, the setting of Problem 1221 (not rechecked here).

Consequence for Problem 1221 (an authored one-line remark): with $\mu_r=\inf_a\limsup_nM_n^r(a)/m_n^r(a)$, Theorem 2 gives $\mu_r\le1+c\log r/r$ for every $r\ge2$ (the printed "for all $r\in\mathbb N$" fails at $r=1$, where it would give $\mu_1\le1$ against the de Bruijn--Erdős value $\mu_1=2$), hence $r(\mu_r-1)\le c\log r$: the third expression of the conjecture can tend to infinity at most at logarithmic rate. Korsky's 2026 preprint claims the matching lower bound $\mu_r-1\ge\log r/(100r)$ for large $r$ (Theorem 1.1 there, claimed and unreviewed), which together with this theorem would fix the order of $\mu_r-1$.

Bears on. Problem 1221: an upper bound on the growth of the third expression; it bears on the rate, not on whether the conjecture holds. The problem page for 480 lists this preprint among the citing records of the 1981 Chung--Graham announcement as a lead only.

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