Abstract Improving a result of Hajela, we show for every function f with limn→∞f(n) = ∞ and f(n) = o(n) that there exists n0 = n0(f) such that for every n ⩾ n0 and any S ⊆ {–1, 1}n with cardinality |S| ⩽ 2n/f(n) one can find orthonormal vectors x1, …, xn ∈ ℝn satisfying ∥ε1x1+⋯+εnxn∥∞⩾clog⁡f(n) $\begin{array}{} \displaystyle \|\varepsilon_1x_1+\dots+\varepsilon_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)} \end{array}$ for all (ε1, …, εn) ∈ S. We obtain analogous results in the case where x1, …, xn are independent random points uniformly distributed in the Euclidean unit ball B2n $\begin{array}{} \displaystyle B_2^n \end{array}$ or in any symmetric convex body, and the ℓ∞n $\begin{array}{} \displaystyle \ell_{\infty }^n \end{array}$-norm is replaced by an arbitrary norm on ℝn.

中文翻译：

---

关于有符号向量和的范数的注记

摘要 改进 Hajela 的结果，我们证明对于每个函数 f，其中 limn→∞f(n) = ∞ 和 f(n) = o(n) 存在 n0 = n0(f) 使得对于每个 n ⩾ n0 和任意 S ⊆ {–1, 1}n 具有基数 |S| ⩽ 2n/f(n) 可以找到正交向量 x1, …, xn ∈ ℝn 满足 ∥ε1x1+⋯+εnxn∥∞⩾clog⁡f(n) $\begin{array}{} \displaystyle \|\varepsilon_1x_1+\dots+ \varepsilon_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)} \end{array}$ for all (ε1, …, εn) ∈ S。我们在 x1, ..., xn 是均匀分布在欧几里德单位球 B2n $\begin{array}{} \displaystyle B_2^n \end{array}$ 或任何对称凸体中的独立随机点，并且 ℓ∞n $\begin{ array}{} \displaystyle \ell_{\infty }^n \end{array}$-norm 被 ℝn 上的任意范数代替。