The existence of unimodular forms with small norms on sequence spaces is crucial in a variety of problems in modern analysis. We prove that the infimum of $\left\Vert A\right\Vert $ over all unimodular $d$-linear (complex or real) forms $A$ on $\ell_{ p_1}^{ n_{ 1}} \times \cdots \times \ell_{ p_d}^{ n_{ d}}$, for all $p_1,\dots,p_d \in [2, \infty]$ and all positive integers $n_1,\dots,n_d$, behaves (asymptotically) as $\left(n_{ 1}^{ 1/2} + \cdots + n_{ d}^{ 1/2}\right) \prod_{ j=1}^{ d}n_{ j}^{ \frac{ 1}{ 2} - \frac{ 1}{ p_j}}$. Applications to the theory of the multilinear Hardy--Littlewood inequality are also presented.
Categories: math.FA