AI 中文总结
该研究针对有限维希尔伯特空间上系数为±1的实多重线性型，确定了K_{r,n}=1的条件，推导了等式不成立时的范数间隙，还给出了渐近结果、四阶矩估计及相关构造。
AI 中文摘要
我们研究有限维希尔伯特空间上系数取自{−1,1}的实多重线性型。ℓ₂ʳ×ℓ₂ⁿ×ℓ₂ⁿ上的每一个三线性符号型的范数至少为√n。记K_{r,n}为最小范数除以√n的值，我们证明当阶数为n的阿达马（Hadamard）矩阵存在且r≤ρ(n)（ρ为Hurwitz–Radon函数）时，K_{r,n}=1。若等式不成立，我们得到一个明确的大于√n的间隙。我们还证明了两个渐近结果：若1≤mₙ≤n且limsup rₙ/log₂n<2，则ℓ₂ʳⁿ×ℓ₂ᵐⁿ×ℓ₂ⁿ上存在符号型，其范数为(1+o(1))√n；在平方情形下，若r≥2⌈log₂(8n)⌉，则K_{r,n}−1≥c(1+log₂n)⁻⁴。我们还对每个固定的多重线性阶数证明了四阶矩估计，刻画了等号成立的情形，并给出了精确与渐近构造。
英文摘要
We study real multilinear forms with coefficients in $\{-1,1\}$ on finite-dimensional Hilbert spaces. Every trilinear sign form on $\ell_2^r\times\ell_2^n\times\ell_2^n$ has norm at least $\sqrt n$. Writing $K_{r,n}$ for the least norm divided by $\sqrt n$, we prove that $K_{r,n}=1$ exactly when a Hadamard matrix of order $n$ exists and $r\leρ(n)$, where $ρ$ is the Hurwitz--Radon function. If equality fails, we obtain an explicit gap above $\sqrt n$. We also prove two asymptotic results. If $1\le m_n\le n$ and $\limsup r_n/\log_2 n<2$, there are sign forms on $\ell_2^{r_n}\times\ell_2^{m_n}\times\ell_2^n$ with norm $(1+o(1))\sqrt n$. In the square case, if $r\ge2\lceil\log_2(8n)\rceil$, then $K_{r,n}-1\ge c(1+\log_2 n)^{-4}$. We also prove a fourth-moment estimate in every fixed multilinear order, characterize equality, and give exact and asymptotic constructions.
