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 $m\ge 2$ and $d\in [1,m]$, we determine the product-summability exponent and the multilinear summability invariant: \[ \mathrm{prod}(m,d) =\min\left\{\frac{m}{d},\,m-\lceil d\rceil+1\right\}, \qquad γ_{\mathrm{mult}}(m,d) =\min\left\{m-\lceil d\rceil+1,\frac{2m}{d+1}\right\}. \] In particular, ${\gamma }_{\mathrm{mult}}(4,2)=8/3$, showing that the multilinear invariant need not be an integer. We also prove that, for every $d\in [1,m]$, 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 $\mathbf{p}=(p_1,\dots ,p_m)$ with ${\sum }_{j}1/p_j<1$, simultaneously across the two regimes separated by ${\sum }_{j}1/p_j=1/2$.

51 pages. This version substantially expands the previous manuscript and includes new results