Polarization of an inequality

TL;DR

This paper extends a key inequality related to exponential sums on the unit circle, using Toeplitz matrix techniques to handle more complex logarithmic expressions involving multiple terms.

Contribution

It introduces a generalized inequality replacing a single logarithmic term with a sum over multiple terms, advancing the understanding of Littlewood conjecture variants.

Findings

Established a new inequality for sums of exponential functions.

Connected the inequality to properties of Toeplitz matrices and their determinants.

Provided a proof framework based on matrix polarization and mixed discriminants.

Abstract

We generalize a previous inequality related to a sharp version of the Littlewood conjecture on the minimal $L_1$-norm of $N$-term exponential sums $f$ on the unit circle. The new result concerns replacing the expression $\log(1+t|f|^2)$ with $\log (\sum_{k=1}^K t_k|f_{k}|^2)$. The proof occurs on the level of finite Toeplitz matrices, where it reduces to an inequality between their polarized determinants (or "mixed discriminants").