A finite victory over de Bruijn-Erdős in interval discrepancy

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 $\text{disc}(n)\leq 2-\frac{3}{2n}+O(1/n^2)$. In this paper, we improve the first-order term of this bound. Specifically, we construct a strategy, called \emph{lex-merge}, with $\text{disc}(n)\leq 2-\frac{4\ln 2}{n}+O(1/n^2)$. We prove also the lower bound $\text{disc}(n)\geq 2-\frac{6\ln 2}{n}-O(1/n^2)$, showing that the first-order term in this improvement over the de Bruijn--Erdős construction has the correct order of magnitude. We conjecture that the lex-merge strategy is optimal for every $n$.