Constants of the Kahane--Salem--Zygmund inequality asymptotically bounded by $1$ II

In [18] we have shown that, for $p_1,p_2\in (2,\mathrm{\infty }]$, the constants of Bennett’s inequality on unimodular bilinear forms on $\ell_{p_{1}}^{n_{1} }\times\ell_{p_{2}}^{n_{2}}$areasymptoticallyboundedby$1$. In the present paper we use a different approximation technique to investigate the remaining cases $p_1,p_2\in [1,\mathrm{\infty }]\mathrm{.}$ This new approach also provides a stronger asymptotic control, in the sense that the constants are “uniformly” asymptotically bounded by $1$, with no dependence on $p_1,p_2\mathrm{.}$