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($\mathrm{Re}f(z)>0$) and left($\mathrm{Re}f(z)<0$) half-planes, respectively. Note that $\mathrm{Re}f(z)$ 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.