Symmetric Kahane--Salem--Zygmund Inequalities and the Supremum Norm

The Kahane–Salem–Zygmund inequality provides unimodular $m$-linear forms with small supremum norm. We consider its dimension-free symmetric constant $C_m^{\mathrm{s}\mathrm{y}\mathrm{m}}$ over the real scalar field, defined by the requirement that, for every $n$, some real symmetric unimodular $m$-linear form $A:({\mathrm{\ell }}_{\mathrm{\infty }}^n{)}^m\to \mathbb{R}$ satisfy $$\mathrm{\parallel }A\mathrm{\parallel }\le C_m^{\mathrm{s}\mathrm{y}\mathrm{m}}{n}^{(m+1)\mathrm{/}2}\mathrm{.}$$ For complex scalars, Boas obtained under permutation symmetry an upper bound of order at most $\sqrt{m\log m}\sqrt{m!}$; over the real scalar field, an elementary argument gives the sharper order $\sqrt{m}\sqrt{m!}$. We prove $$C_m^{\mathrm{s}\mathrm{y}\mathrm{m}}\ge (\sqrt{\frac{2}{e}}+o(1))\frac{\sqrt{m!}}{m},$$ using the square-free Walsh spectrum of the diagonal polynomial. In the opposite direction, we establish $$C_m^{\mathrm{s}\mathrm{y}\mathrm{m}}\le C_0\sqrt{m!},$$ where $C_0$ is absolute, removing the factor $\sqrt{m}$ from the real upper bound. This estimate follows from a geometric argument in which rigidity for Gram permanents reduces the relevant configurations to sets controlled by Gaussian width. For unrestricted unimodular forms, we also obtain a quantitative rectangular estimate from truncated Hadamard matrices; in equal dimension, the normalized minimum is at most $1+o(1)$ whenever $m=o(n^{5\mathrm{/}6})$.