  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 $D$. 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, almost periodic functions,
functions of several variables and multivalued algebraic functions.
