Here is a deep theorem about linear recursive sequences:

Theorem 4 (Mahler-Lech theorem)
Let be a linear recursive sequence
of complex numbers, and let be a complex number.
Then there exists a finite (possibly empty) list of arithmetic progressions
, , ...
and a finite (possibly empty) set of integers
such that

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