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For a finite set of natural numbers $D$ consider a complex polynomial of theform $f(z) = \sum_{d \in D} c_d z^d$. Let $\rho_+(f)$ and $\rho_-(f)$ be thefractions 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, whoseallowed set of frequencies is $D$. Turns out that $\min(\rho_+(f), \rho_-(f))$is always bounded below by a numerical characteristic $\alpha(D)$ of our set$D$ that arises from a seemingly unrelated combinatorial problem. Furthermore,this result could be generalized to power series, functions of severalvariables and multivalued algebraic functions.

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