通过二元细化的最优有限区间偏差
Optimal Finite Interval Discrepancy via Binary Refinement
浏览论文内容
中文总结 AI 辅助
DeLeo等人提出有限时间de Bruijn-Erdős区间偏差问题并构造lex-merge策略，推测其偏差值最优，本文证明该猜想并建立正质量二元细化过程的严格下界，同时给出r元细化的通用下界。
中文摘要 AI 辅助
DeLeo、Henderschedt和Wells引入了经典de Bruijn-Erdős区间偏差问题的有限时间版本。从单位区间出发，人们反复将现有区间拆分为两个，直到出现n个区间，且要最小化所有中间划分中最长区间与最短区间的最大比值。他们构造了lex-merge策略，其偏差为2^(1-1/⌈n/2⌉)，并推测该值对每个n都是最优的。我们证明了这一猜想。更一般地，我们为正质量的任意二元细化过程建立了一个严格下界：任何从一个正质量出发、反复将一个质量替换为两个总质量相同的正质量、最终得到n个质量的过程，在某个阶段必然存在最大质量与最小质量的比值至少为2^(1-1/⌈n/2⌉)。该证明跟踪了细化过程中的最小质量，并利用了过程中点附近一块质量的强制留存。我们还记录了r元细化对应的通用下界。
英文摘要
DeLeo, Henderschedt, and Wells introduced a finite-horizon version of the classical de Bruijn--Erdos interval discrepancy problem. Starting from the unit interval, one repeatedly splits an existing interval into two until $n$ intervals are present, and one minimizes the largest ratio between the longest and shortest intervals over all intermediate partitions. They constructed the lex-merge strategy, whose discrepancy is $2^{1-1/\lceil n/2\rceil}$, and conjectured that this value is optimal for every $n$. We prove the conjecture. More generally, we establish a sharp lower bound for arbitrary binary refinement processes of positive masses: any process that starts with one positive mass, repeatedly replaces one mass by two positive masses with the same total, and terminates with $n$ masses must at some stage have largest-to-smallest ratio at least $2^{1-1/\lceil n/2\rceil}$. The proof tracks the minimum mass under refinement and uses the forced survival of a piece near the midpoint of the process. We also record the corresponding universal lower bound for $r$-ary refinements.
