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

Abstract: The Kahane--Salem--Zygmund inequality provides unimodular $m$-linear forms with small supremum norm. We consider its dimension-free symmetric constant $C_m<sup>{\mathrm{sym}}$ over the real scalar field, defined by the requirement that, for every $n$, some real symmetric unimodular $m$-linear form $$A:(\ell_\infty^{n)^m\to\mathbb} R$$ satisfy [ |A|\le C_m^{\mathrm{sym}}n^{(m+1)/2}. ] 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{sym}}\ge \left(\sqrt{\frac2e}+o(1)\right)\frac{\sqrt{m!}}m, ] using the square-free Walsh spectrum of the diagonal polynomial. In the opposite direction, we establish [ C_m^{\mathrm{sym}}\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 $m=o(n<sup>{5/6})$.