Density of Lacunary Functions

I'm going to prove density in compact-open topology, i.e. that for any $f$ holomorphic on the disk there is a sequence of lacunary functions convergent to $f$ uniformly on compact sets.

Take any function $g$ lacunary in the unit disk satisfying $g(0)=0$. Then it's easy to see that for $g_n(x)=g(x^n)$, $g_n(x)$ is lacunary again and $g_n$ converges to $0$ uniformly on compacts. Let $f_n$ be the $n$-th Taylor polynomial of $f$, then $f_n$ converges to $f$ uniformly on compacts. It follows $f_n+g_n$ is a sequence of lacunary functions convergent to $f$ uniformly on compacts.

Much more is true: the "non-lacunary" series are very rare, from the point of view of Baire's cathegory, in various natural topologies, and from the point of view of probability. These results go back to Polya and Hausdorff, and a nice survey of them is contained in chapter 4 of the book of Bieberbach, Analytische Fortsetzung, Springer 1955.