The Mahler-Lech theorem

Here is a deep theorem about linear recursive sequences:

Theorem 4 (Mahler-Lech theorem)   Let $\{a_n\}$ be a linear recursive sequence of complex numbers, and let $c$ be a complex number. Then there exists a finite (possibly empty) list of arithmetic progressions $T_1$, $T_2$, ...$T_m$ and a finite (possibly empty) set $S$ of integers such that

$$\{ n \mid a_n=c \} = S \cup T_1 \cup T_2 \cup \cdots \cup T_m.$$

Warning: don't try to prove this at home! This is very hard to prove. The proof uses ``$p$-adic numbers.''