Coarse equidistribution of the argument of entire functions of finite order

Анотація

Given a sector $S\subset \mathbb{C}$ and an entire function $f$ of order $\rho$, we estimate from below the relative area of the preimage $f^{-1}S$. We show that there exist arbitrarily large $r$ such that for any sector $S$ of opening $\alpha$, the relative area of the set $\{f^{-1}S\}\cap r\mathbb{D}$ is bounded from below by $\alpha \cdot\kappa(\rho)$, where $\kappa(\rho)>0$  depends only on $\rho$, and $\kappa(\rho)\sim \textrm{ const } \rho^{-1}$ for $\rho\to\infty$.