Komlós猜想在复差异中的研究
The Komlós conjecture for complex discrepancy
- Courant Institute, New York University（纽约大学库朗数学科学研究所）
- Department of Mathematics, The Ohio State University（俄亥俄州立大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明了Komlós猜想在允许符号取单位模复数（即复差异）时成立，得到有限显式常数，解决了高斯差异的Komlós问题，并利用Burkholder和Bellman函数方法。
AI中文摘要：
Komlós猜想是差异理论中的一个经典问题；它询问是否存在一个绝对常数$K$，使得给定任意$n$个位于$m$维欧几里得球内的向量$a_1,\ldots,a_n$，无论$m,n$多大，总存在符号$\u03b5_1,\ldots,\u03b5_n$的选择，保证$$\u2016\u03b5_1a_1+\u2026+\u03b5_na_n\u2016_\infty \leq K.$$我们证明，如果允许$\varepsilon_i$不仅取$\pm 1$的值，还可以取任何单位模复数（我们称之为复差异），那么上述不等式对于有限的显式常数$K_{\mathbb{C}}$成立。这里，$\mathbb{C}^m$中所得向量的$\ell^\infty$范数为其条目模的最大值，因此实向量的复差异等价于它们的秩-$2$向量差异。因此，我们的结果解决了高斯差异的Komlós问题——这是由Chewi、Gerber、Rigollet和Turner引入的一种差异度量。我们的论文建立在Bansal和Jiang最近关于Beck-Fiala和Komlós猜想的工作之上，我们从Burkholder的形式主义以及概率和调和分析中的Bellman函数方法入手。我们的工作部分源于认识到任何酉矩阵的列的复差异等于1，这一事实基于Idel和Wolf对酉矩阵Sinkhorn正规形的推广的简单计算得出。
英文摘要：
The Komlós conjecture is a classic problem in discrepancy theory; it asks whether an absolute constant $K$ exists such that given any $n$ vectors $a_1,\ldots,a_n$ inside the $m$-dimensional Euclidean ball, regardless of how large $m,n$ are, there is always a selection of signs $\varepsilon_1,\ldots,\varepsilon_n$ guaranteeing $$\|\varepsilon_1a_1+\ldots+\varepsilon_na_n\|_\infty \leq K.$$ We show that if the $\varepsilon_i$'s are allowed to take not just the values of $\pm 1$ but any unit modulus complex number, which we refer to as complex discrepancy, then the above inequality holds for a finite, explicit constant $K_{\mathbb{C}}$. Here, the $\ell^\infty$ norm of the resulting vector in $\mathbb{C}^m$ is the largest modulus of its entries, and thus the complex discrepancy of real vectors is equivalent to their rank-$2$ vector discrepancy. This quantity provides an upper bound (modulo uniform constant prefactor) on Gaussian discrepancy -- a discrepancy measure introduced by Chewi, Gerber, Rigollet and Turner. Thus, we also resolve the Komlós conjecture for Gaussian discrepancy. Our paper builds upon the recent work of Bansal and Jiang on the Beck-Fiala and Komlós conjectures, which we approach from the formalism of Burkholder and the Bellman function method from probability and harmonic analysis. Our work was in part motivated by the realization that the complex discrepancy of the columns of any unitary matrix is equal to 1, a fact that follows from a straightforward calculation based on Idel and Wolf's generalization of the Sinkhorn normal form for unitary matrices.
