SCHWEITZER 1968 Problem 5

Let $k$ be a positive integer, $z$ a complex number, and $\varepsilon <\frac12$ a positive number. Prove that the following inequality holds for infinitely many positive integers $n$ : $$\mid \sum_{0\leq l \leq \frac{n}{k+1}} \binom{n-kl}{l}z^l \mid \geq (\frac 12-\varepsilon)^n.$$

P. Turan