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Source. F. Clément and S. Steinerberger, Balanced stick breaking, arXiv:2511.14637v1, Theorem 2 and Theorem 3 (p. 2), the remark after Theorem 3 (p. 3) and the paragraph "Theorem 3 implies Theorem 2" at the end of Section 2 (p. 9) of the retained PDF, read in the canonical conversion and checked against the text layer; held by its library card, Clément and Steinerberger 2025, with the result page Theorem 2.

Standing. Author-recorded reconstruction of the derivation of Theorem 2 from Theorem 3; not an independent review; changes no status and assigns no tier. The source is an unrefereed preprint. Theorem 3 is imported from the same source as a same-paper input whose proofs (Section 2 for the van der Corput sequence, pp. 3--9; Section 3 for the golden-ratio Kronecker sequence, pp. 9--11) are not reconstructed.

Definitions

A sequence $(x_k)_{k\ge1}$ on the circle $S^1\cong[0,1)$ of length $1$; its first $n$ terms cut the circle into $n$ arcs (the source's abstract and introduction count $n+1$ pieces of the circular stick $S^1$ after $n$ breaks, and its Section 2 sets $x_0=0$, so the source treats $0$ as an extra break point; here $0$ is not a break point, the convention of Problem 1221). For $n$ points $y_1,\ldots,y_n$ in cyclic order, an $r$-span is the closed arc $[y_i,y_{i+r}]$ from a point to the point $r$ places later; for $r+1\le n$ it contains exactly the $r+1$ points $y_i,\ldots,y_{i+r}$ of the first $n$ terms. The two sequences are the base-$2$ van der Corput sequence $\frac12,\frac14,\frac34,\frac18,\ldots$ and the Kronecker sequence $x_k=\{k\varphi\}$ with $\varphi=(1+\sqrt5)/2$.

The imported input (Theorem 3, p. 2, in its circle form)

Let $(x_k)$ be either sequence. There is a universal constant $c>0$ such that for every $r\in\mathbb N$ there is $n_0(r)$ with the following property: for every $n\ge n_0(r)$ and every closed arc $I$ of $S^1$ of length $r/n$,

$$\Bigl|\#\{1\le k\le n:\ x_k\in I\}-r\Bigr|\ \le\ c\log r .$$

The printed statement takes $I=[x,x+r/n]$ with $0\le x\le1-r/n$; the sentence after it says the stronger form for all arcs of length $r/n$ on $S^1$ is what is actually shown, and that form is what the derivation uses. See "Boundary at $r=1$" below.

Statement (Theorem 2, p. 2)

There exist a sequence in $[0,1]\cong S^1$ (either of the two above) and a universal constant $c'<\infty$ such that for every integer $r\ge2+c\log2$, with $c$ the constant of Theorem 3, and every $n\ge n_1(r)$, the first $n$ terms satisfy

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

The source states "for all $r\in\mathbb N$"; the restriction to $r\ge2$ is explained below, and the further restriction to $r\ge2+c\log2$ is the range on which the derivation from Theorem 3 goes through (see the proof and the Source notes).

Proof

Let $c$ be the constant of Theorem 3 and fix an integer $r\ge2+c\log2$. Define

$$R^+=\min\{R\in\mathbb N:\ R-c\log R\ge r+2\},\qquad R^-=\max\{R\in\mathbb N:\ R\ge2,\ R+c\log R\le r\};$$

$R^-$ exists because $R=2$ qualifies when $r\ge2+c\log2$, and $R^-\ge2$ keeps Theorem 3 at $R^-$ inside the range $R\ge2$ of the import, and $R^+$ exists because $R-c\log R\to\infty$. Take $n\ge\max(n_0(R^+),n_0(R^-),R^++1)$, so that arcs of length $R^+/n$ are shorter than the circle and $r+1<n$.

Every $r$-span is shorter than $R^+/n$. Let $J=[y_i,y_{i+r}]$ be an $r$-span and suppose $|J|\ge R^+/n$. Then $J$ contains a closed arc $I$ of length exactly $R^+/n$, and Theorem 3 at $R^+$ gives $\#\{k\le n:x_k\in I\}\ge R^+-c\log R^+\ge r+2$; but $I\subseteq J$ and $J$ contains exactly $r+1$ of the first $n$ terms. So $|J|<R^+/n$.

Every $r$-span is longer than $R^-/n$. Suppose $|J|\le R^-/n$. Then $J$ lies inside a closed arc $I$ of length exactly $R^-/n$, and Theorem 3 at $R^-$ gives $\#\{k\le n:x_k\in I\}\le R^-+c\log R^-\le r$; but $I$ contains the $r+1$ terms of $J$. So $|J|>R^-/n$.

The ratio. Hence, for every $n\ge n_1(r):=\max(n_0(R^+),n_0(R^-),R^++1)$,

$$\frac{\text{largest $r$-span}}{\text{smallest $r$-span}}\ <\ \frac{R^+}{R^-}.$$

Asymptotics of $R^\pm$. Let $x_0=\lceil r+2+2c\log r\rceil$. For $r\ge r_1(c)$ we have $r+3+2c\log r\le r^2$, so $c\log x_0\le2c\log r$ and $x_0-c\log x_0\ge r+2$; as $R^+$ is the least such integer, $R^+\le x_0\le r+3+2c\log r$. Let $x_1=\lfloor r-c\log r\rfloor$, which is at least $1$ for $r\ge r_1(c)$; then $x_1\le r$ gives $x_1+c\log x_1\le r$, so $R^-\ge x_1\ge r-c\log r-1$. For $r\ge r_1(c)$ large enough that $r-c\log r-1\ge r/2$,

$$\frac{R^+}{R^-}\ \le\ 1+\frac{4+3c\log r}{r-c\log r-1}\ \le\ 1+\frac{2(4+3c\log r)}r\ \le\ 1+\frac{(6c+12)\log r}r ,$$

using $4\le6\log2\le6\log r$ for $r\ge2$. For $2+c\log2\le r<r_1(c)$ the ratio is at most $R^+(r)/2\le R^+(r_1)$, a constant, while $\log r/r\ge\log2/r_1$; so a single universal $c'$, at least $6c+12$ and at least $R^+(r_1)r_1/\log2$, gives the statement for every $r\ge2+c\log2$.

Source notes

• Boundary at $r=1$. With $\log1=0$, Theorem 3 at $r=1$ would say that every closed arc of length $1/n$ contains exactly one of the first $n$ terms, and Theorem 2 at $r=1$ that all $n$ gaps are equal, for all large $n$. Both fail for both sequences: for the van der Corput sequence at $n=2^k-1$ the first $n$ terms are the points $j/2^k$, $1\le j\le2^k-1$, the closed arc $[1/2^k,1/2^k+1/n]$ contains two of them, and the gap through $0$ is twice the others; for the Kronecker sequence the $n$ gaps are never all equal, since equal spacing $1/n$ would make $\{\varphi\}$ rational. (An authored observation, checked here.) The statements are to be read for $r\ge2$, or with $\log r$ replaced by $\log(2r)$; the derivation above is for $r\ge2+c\log2$, and the boundary is immaterial to the growth question.
• The range $2\le r<2+c\log2$ is not derived. For such $r$ no integer $R\ge2$ has $R+c\log R\le r$, and Theorem 3 at $R\ge2$ alone gives no lower bound on the smallest $r$-span: when $r+1\le c\log2$, $n-r$ equally spaced points with $r$ further points clustered next to one of them satisfy every Theorem 3 inequality at $R\ge2$ (a closed arc of length $R/n$ holds between $R-1$ and $R+1$ grid points once $n\ge Rr$, plus at most $r$ cluster points) while one $r$-span is arbitrarily short; the source gives no value of $c$. The source asserts Theorem 2 for all $r\in\mathbb N$; on this finite range its bound needs an input not imported here, such as a minimum-gap bound for each sequence.
• A label slip on p. 9. The paragraph deriving Theorem 2 twice says "Theorem 2 implies" where Theorem 3 is meant (checked in the text layer).
• The paragraph on p. 9 states the two bounds as "cannot be shorter than $r-c\log r$" and "cannot be longer than $r+c\log r$" without the integer thresholds $R^\pm$ or the dependence of $n$ on $r$; the definitions of $R^\pm$ and the threshold $n_1(r)$ are the corpus's completion.

Reading addressed

The theorem bounds the third constant from above: with $\mu_r=\inf_a\limsup_nM_n^r(a)/m_n^r(a)$ over all sequences (either witness is a sequence of distinct points, so the same bound holds over the distinct-point family),

$$\mu_r\ \le\ 1+\frac{c'\log r}r,\qquad r(\mu_r-1)\ \le\ c'\log r\qquad(r\ge2+c\log2),$$

the same under both readings of Problem 1221 since the third expression needs no normalization. It bears on the rate, not on whether $r(\mu_r-1)\to\infty$; the matching claimed lower bound $\mu_r-1\ge\log r/(100r)$ is the ratio part of Korsky's Theorem 1.1.