Year: 2026
Type: preprint
Abstract: We study a finite form of the classical interval discrepancy problem. Starting from the unit interval, one repeatedly splits an existing interval into two until $n$ intervals have been produced. The discrepancy of such a process is the maximum, over all intermediate stages, of the ratio between the longest interval and the shortest interval. A theorem of de Bruijn and Erdős from 1949 shows that this ratio must approach $2$ as $n\to\infty$, and they give a sharp construction achieving this bound. For fixed $n$, their construction gives the upper bound $\operatorname{disc}(n)\leq 2-\frac{3}{2n}+O\bigl(\frac 1{n^2}\bigr)$. In this paper, we prove that $\operatorname{disc}(n)=2^{1-1/\lceil n/2\rceil}=2-\frac{4\ln 2}{n}+O\bigl(\frac 1{n^2}\bigr)$ for every $n$.
Source: arXiv (Cornell University)
Authors: DeLeo, Jared, Henderschedt, Owen, Wells, Chris
Institutions: La Salle University, Auburn University, Haverford College
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162806100