A question around Liouville's Theorem

Liouville's theorem of complex analysis states that a bounded entire function is constant. This means that for a non-constant entire function $f$ there exists a sequence $z_n \in \mathbb{C}$ such that $|z_n| \to \infty$ and $f(z_n) \to \infty$. I am trying to understand if a sort of converse holds in the following sense: consider a closed set $S \subset \mathbb{C}$ ($S$ may be unbounded) such that $\mathbb{C} \setminus S$ is unbounded. Can one construct a non-constant entire function $f$ such that $f$ is bounded on $S$ ?

Comment: Cross-posted on MO here.

• Try the Maximum modulus principle. But see math.stackexchange.com/questions/894304/…. – lhf Oct 28 '16 at 17:14
• What if $S= \{0\}?$ – zhw. Oct 28 '16 at 17:21
• @lhf What if the maximum is attained at some boundary point of $S$? – user346998 Oct 28 '16 at 17:53
• Yes, for every curve $\gamma$ such that $\gamma(t) \to \infty$, there exists a function bounded everywhere except in the neighborhood of the curve, search for the Alice Roth's theorems on entire functions – reuns Oct 30 '16 at 9:49