Lower bounds on the measure of the support of positive and negative parts of trigonometric polynomials

TL;DR

This paper establishes lower bounds on the measure of the support of positive and negative parts of real parts of trigonometric polynomials with frequencies from a finite set, linking it to a combinatorial characteristic of that set.

Contribution

It introduces a new lower bound for the support measure of the positive and negative parts of such polynomials, connecting harmonic analysis with combinatorial properties of frequency sets.

Findings

Lower bounds depend on a combinatorial characteristic of the frequency set

Results extend to power series, almost periodic functions, and multivariable functions

Supports of positive/negative parts are quantitatively constrained by the set D

Abstract

For a finite set of natural numbers $D$ consider a complex polynomial of the form $f(z) = \sum_{d \in D} c_d z^d$. Let $\rho_+(f)$ and $\rho_-(f)$ be the fractions of the unit circle that $f$ sends to the right($\operatorname{Re} f(z) > 0$) and left($\operatorname{Re} f(z) < 0$) half-planes, respectively. Note that $\operatorname{Re} f(z)$ is a real trigonometric polynomial, whose allowed set of frequencies is $D$. It turns out that $\min(\rho_+(f), \rho_-(f))$ is always bounded from below by a numerical characteristic $\alpha(D)$ of our set $D$ which comes from a seemingly unrelated combinatorial problem. Furthermore, this result could be generalized to power series, almost periodic functions, functions of several variables and multivalued algebraic functions.