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Sharp Summability on Supports of Prescribed Combinatorial Dimension
Abstract
We solve four questions raised by Bayart concerning coefficient summability for multilinear forms with prescribed supports. For every m≥ 2 and d∈[1,m], we determine the product-summability exponent and the multilinear summability invariant: prod(m,d) =min{m/d, m−⌈ d⌉+1}, γmult(m,d) =min{m−⌈ d⌉+1,2m/(d+1)}. In particular, γmult(4,2)=8/3, showing that the multilinear invariant need not be an integer. We also prove that, for every d∈[1,m], there is a single infinite support of exact combinatorial dimension d on which the dimensional Hardy--Littlewood bound is attained over both scalar fields for every anisotropic parameter p=(p1,…,pm) with ∑j 1/pj<1, simultaneously across the two regimes separated by ∑j 1/pj=1/2.
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BibTeX
@misc{barbosa2026sharp,
title = {{Sharp Summability on Supports of Prescribed Combinatorial Dimension}},
author = {Anderson Barbosa and Daniel Pellegrino and Anselmo Raposo Jr and Eduardo Teixeira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06847},
eprint = {2609.06847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06847},
}
RIS
TY - UNPB
TI - Sharp Summability on Supports of Prescribed Combinatorial Dimension
AU - Anderson Barbosa
AU - Daniel Pellegrino
AU - Anselmo Raposo Jr
AU - Eduardo Teixeira
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.06847
UR - https://arxiv.org/abs/2609.06847
AB - We solve four questions raised by Bayart concerning coefficient summability for multilinear forms with prescribed supports. For every m≥ 2 and d∈[1,m], we determine the product-summability exponent and the multilinear summability invariant: prod(m,d) =min{m/d, m−⌈ d⌉+1}, γ_mult(m,d) =min{m−⌈ d⌉+1,2m/(d+1)}. In particular, γ_mult(4,2)=8/3, showing that the multilinear invariant need not be an integer. We also prove that, for every d∈[1,m], there is a single infinite support of exact combinatorial dimension d on which the dimensional Hardy--Littlewood bound is attained over both scalar fields for every anisotropic parameter p=(p_1,…,p_m) with ∑_j 1/p_j<1, simultaneously across the two regimes separated by ∑_j 1/p_j=1/2.
ER -