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

TL;DR

This paper extends previous work on the Kahane--Salem--Zygmund inequality, showing that the associated constants are asymptotically bounded by 1 across all p-values, using a new approximation technique for broader cases.

Contribution

Introduces a novel approximation method to establish uniform asymptotic bounds for constants in the inequality across all p-values.

Findings

Constants are asymptotically bounded by 1 for all p-values.

New technique provides stronger, uniform asymptotic control.

Extends previous results to remaining cases p1,p2 in [1,∞].

Abstract

In [18] we have shown that, for $p_{1},p_{2}\in(2,\infty]$, the constants of Bennett's inequality on unimodular bilinear forms on $\ell_{p_{1}}^{n_{1} }\times\ell_{p_{2}}^{n_{2}}$ are asymptotically bounded by $1$. In the present paper we use a different approximation technique to investigate the remaining cases $p_{1},p_{2}\in\lbrack1,\infty].$ 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 $p_{1},p_{2}.$