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

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.