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Statement

For a sequence $a=(a_1,a_2,\ldots)$ of numbers mod $1$, the points $a_1,\ldots,a_n$ cut the circle of circumference $1$ into $n$ intervals; $M_n^1(a)$ and $m_n^1(a)$ are the largest and smallest of their lengths, $\Lambda_1(a)=\limsup_n nM_n^1(a)$, $\lambda_1(a)=\liminf_n nm_n^1(a)$, $\mu_1(a)=\limsup_n M_n^1(a)/m_n^1(a)$, and $\Lambda_1$, $\lambda_1$, $\mu_1$ are the infimum, the supremum and the infimum of these over all sequences (Section 1, p. 14).

The $r=1$ values (p. 14, proved in Section 2 and Sections 3--5).

$$\Lambda_1=\frac1{\log2},\qquad \lambda_1=\frac1{\log4},\qquad \mu_1=2 .$$

The sequence $a_k=\log_2(2k-1)$ reduced mod $1$ attains all three: $\Lambda_1(a)=1/\log2$, $\lambda_1(a)=1/\log4$, $\mu_1(a)=2$ (p. 15).

Source. N. G. de Bruijn and P. Erdős, Sequences of points on a circle, Proc. 52 (1949), 14--17; the values on printed p. 14 (PDF p. 2 of the TU/e portal PDF), Section 2 on pp. 14--15 (PDF pp. 2--3), read on the page images. The edition read is identified in the source digest.

Read depth. Claims checked: the definitions, the displayed values and the Section 2 computation were read clause by clause on the page images; the matching universal bounds are the $r=1$ cases of the Section 3--5 results, whose proofs were read for structure only.

Proof pointer

Section 2 shows that $a_1,\ldots,a_n$ sit on the circle in the same cyclic order as $\log_2n,\log_2(n+1),\ldots,\log_2(2n-1)$, display (2.1), because reduction mod $1$ pairs the $a_k$ with $k\le n$ one-to-one with these numbers, the residues within each list being pairwise distinct. The $n$ intervals therefore have lengths $\log_2\frac{n+1}n,\log_2\frac{n+2}{n+1},\ldots,\log_2\frac{2n}{2n-1}$ (the last one wrapping around), so

$$nM_n^1(a)=\frac{n\log(1+1/n)}{\log2},\qquad nm_n^1(a)=\frac{n\log\bigl(1-\tfrac1{2n}\bigr)^{-1}}{\log2}.$$

As $n\to\infty$ the first increases to $1/\log2$, the second decreases to $1/\log4$, and the ratio $M_n^1(a)/m_n^1(a)$ increases to $2$. The values are best possible because Section 3 gives $\Lambda_1(a')\ge1/\log2$, Section 4 gives $\lambda_1(a')\le1/\log4$ and Section 5 gives $\mu_1(a')\ge2$ for every sequence $a'$ (the $r=1$ cases of the Section 3 bound, (4.3) and (5.7)).

Dependencies

Elementary properties of the logarithm; the universal bounds of Sections 3--5 for sharpness.

Bears on

• Problem 1221: the site's commentary quotes these three values and the witness $a_k=\log_2(2k-1)$; the $r=1$ case of the problem's constants is exactly determined, and the question concerns the growth of the deviations for large $r$.
• Problem 480: background. The Chung--Graham chapter behind that problem attributes to this note the constant $1/\log4$ for its clustering measure of a sequence in the unit interval; that constant is $\lambda_1$ here, on the circle.