A Complex Analogue of Spencer's Six Standard Deviations Theorem and the Complex Banach-Mazur Distance

TL;DR

The paper develops a complex analogue of Spencer's Six Standard Deviations Theorem, proving that for vectors $a_1,\dots,a_n\in \mathbb{C}^n$ with $\|a_i\|_{\infty}\le 1$ there exists a unimodular $x\in \mathbb{C}^n$ with $|\langle x,a_i\rangle|\le \sqrt{n}$ for all $i$, with equality realized by rows of complex Hadamard matrices. It connects this bound to the Banach–Mazur distance between complex $\ell_1^n$ and $\ell_\infty^n$ spaces, showing $d_{BM}(\ell_1^n(\mathbb{C}), \ell_\infty^n(\mathbb{C}))=\sqrt{n}$ if the conjecture holds, and proves the conjecture in dimensions $n=2,3$. The paper also develops a broader conjectural framework for BM-distances between complex $\ell_p^n$ and $\ell_q^n$ spaces, verified here for $n=2$, and studies complex Hadamard matrices as a key tool, illuminating sharpness via the $\sqrt{n}$ bound. The core contribution in 3D is a torus-covering argument using toric bodies and a $9$-point grid to establish Conjecture 1 for $n=3$, highlighting the deep interplay between complex geometry, Hadamard structures, and discrepancy-type results in finite dimensions.

Abstract

We investigate a complex analogue of Spencer's Six Standard Deviations Theorem. Specifically, we propose the following conjecture: for any dimension $n \geq 2$, given vectors $a_1, \ldots, a_n \in \mathbb{C}^n$ satisfying $\|a_i\|_{\infty} \leq 1$ for each $i=1, \ldots, n$, there exists a vector $x \in \mathbb{C}^n$ with all coordinates of modulus one such that $|\langle x, a_i \rangle| \leq \sqrt{n}$ for every $i=1, \ldots, n$. The bound of $\sqrt{n}$ is sharp, as demonstrated by the row vectors of any complex $n \times n$ Hadamard matrix. Furthermore, if the conjecture holds in dimension $n$, it implies that the Banach--Mazur distance between the complex $\ell_1^n$ and $\ell_{\infty}^n$ spaces is equal to $\sqrt{n}$. We prove the conjecture for $n =2, 3$, thereby establishing also that $d_{BM}(\ell_1^n, \ell_{\infty}^n) = \sqrt{n}$ for these dimensions. Additionally, we propose a conjecture about the Banach--Mazur distances between complex $\ell_p^n$ spaces and we verify it for $n=2$. This leads to a complete determination of all possible Banach--Mazur distances between complex $\ell_p^2$ spaces.

Figures (6)

• Figure 1: Visualization of the grid $\mathcal{G}$ and three closed toric bodies centered at the rows of the DFT matrix, that cover $\mathbb{T}$. Black rectangle represents the fundamental domain $[-\pi, \pi)^2$.
• Figure 2: Comparison of toric bodies for different values of $|t|$.
• Figure 3: Covering of the torus by three strip-like closed toric bodies $\overline{B(a_j)}$ where $a_j = (1, \omega^j, 0)$ for $j=1,2,3$.
• Figure 4: A considered subset of Class $3$.
• Figure 5: A considered subset of Class $4$.

Theorems & Definitions (24)

• Theorem 1.1: Spencer's Six Standard Deviations Theorem
• Conjecture 1.2
• Conjecture 1.3
• Conjecture 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Remark 2.2
• proof : Proof of Conjecture \ref{['conj1']} for $n=2$.
• Lemma 3.1