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 Cmsym over the real scalar field, defined by the requirement that, for every n, some real symmetric unimodular m-linear form A:(ℓ∞n)m→ℝ satisfy |A|≤ Cmsymn(m+1)/2. For complex scalars, Boas obtained under permutation symmetry an upper bound of order at most √(mlog m) √(m!); over the real scalar field, an elementary argument gives the sharper order √m √(m!). We prove Cmsym≥ (√(2/e)+o(1))√(m!)/m, using the square-free Walsh spectrum of the diagonal polynomial. In the opposite direction, we establish Cmsym≤ C0√(m!), where C0 is absolute, removing the factor √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(n5/6).
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BibTeX
@misc{pellegrino2026symmetric,
title = {{Symmetric Kahane--Salem--Zygmund Inequalities and the Supremum Norm}},
author = {Daniel M. Pellegrino and Anselmo B. Raposo Jr. and Eduardo V. Teixeira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30663},
eprint = {2609.30663},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30663},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symmetric Kahane--Salem--Zygmund Inequalities and the Supremum Norm
AU - Daniel M. Pellegrino
AU - Anselmo B. Raposo Jr.
AU - Eduardo V. Teixeira
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30663
UR - https://arxiv.org/abs/2609.30663
AB - The Kahane--Salem--Zygmund inequality provides unimodular m-linear forms with small supremum norm. We consider its dimension-free symmetric constant C_m^sym over the real scalar field, defined by the requirement that, for every n, some real symmetric unimodular m-linear form A:(ℓ_∞^n)^m→ℝ satisfy |A|≤ C_m^symn^(m+1)/2. For complex scalars, Boas obtained under permutation symmetry an upper bound of order at most √(mlog m) √(m!); over the real scalar field, an elementary argument gives the sharper order √m √(m!). We prove C_m^sym≥ (√(2/e)+o(1))√(m!)/m, using the square-free Walsh spectrum of the diagonal polynomial. In the opposite direction, we establish C_m^sym≤ C_0√(m!), where C_0 is absolute, removing the factor √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^5/6).
ER -