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Statement

With the Section 1 constants $\Lambda_r$, $\lambda_r$, $\mu_r$ (the infimum over sequences of $\limsup_nnM_n^r(a)$, the supremum of $\liminf_nnm_n^r(a)$ and the infimum of $\limsup_nM_n^r(a)/m_n^r(a)$, where $M_n^r(a)$ and $m_n^r(a)$ are the largest and smallest sums of $r$ consecutive intervals cut by $a_1,\ldots,a_n$ on the circle of circumference $1$), Section 6 (p. 17) says that the bounds (3.3), (4.3) and (5.7) are "probably not best possible if $r\ge2$" and conjectures that the expressions

$$r(\Lambda_r-1),\qquad r(1-\lambda_r),\qquad r(\mu_r-1)$$

"tend to infinity if $r\to\infty$". The introduction (p. 14) states the third part alone, as the conjecture that $r(\mu_r-1)$ is unbounded, and adds that the "just distributions" theorem of van Aardenne-Ehrenfest (Proc. 48 (1945), 266--271 = Indag. Math. 7 (1946), 71--76) would follow from it. The site's Problem 1221 reproduces the Section 6 wording.

Source. N. G. de Bruijn and P. Erdős, Sequences of points on a circle, Proc. 52 (1949), 14--17; Section 6 on printed p. 17 and the remark on p. 14 (PDF pp. 5 and 2 of the TU/e portal PDF), read on the page images. The edition read is identified in the source digest.

Read depth. Claims checked: the two passages were read clause by clause on the page images. A conjecture has no proof to check.

The normalization defect

As worded, the first two expressions are missing a factor of $r$: the average $r$-span is $r/n$, so $\Lambda_r$ and $\lambda_r$ are compared with $r$ rather than with $1$. The community database marks the site's statement as needing this correction ("ambiguous statement"; a missing factor of $r$), and Korsky's 2026 preprint restates the conjecture as $\bar A_r-r\to\infty$, $r-\underline A_r\to\infty$, $r(\mu_r-1)\to\infty$, a corrected paraphrase rather than the note's wording.

Dependencies

None.

Bears on

• Problem 1221: this passage is the problem. The site's wording inherits the normalization slip; the problem page shows the mean-normalized form as its corrected Statement and records the 2026 preprint that claims all three parts of it.