Asymptotic estimates for unimodular multilinear forms with small norms on sequence spaces

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 $\parallel A\parallel$ over all unimodular $d$-linear (complex or real) forms $A$ on ${\mathrm{\ell }}_{p_1}^{n_1}\times \cdots \times {\mathrm{\ell }}_{p_d}^{n_d}$, for all $p_1,\dots ,p_d\in [2,\mathrm{\infty }]$ 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.