Mahler-Lech Theorem

Let $K$ be a Field of Characteristic 0 (e.g., the rationals $\Bbb{Q}$) and let $\{u_n\}$ be a Sequence of elements of $K$ which satisfies a difference equation of the form

$$0 = c_0 u_n + c_1 u_{n+1} +\ldots + c_k u_{n+k},$$

where the Coefficients $c_i$ are fixed elements of $K$. Then, for any $c \in K$, we have either $u_n = c$ for only finitely many values of $n$, or $u_n = c$ for the values of $n$ in some Arithmetic Progression.

The proof involves embedding certain fields inside the p-adic Number $\Bbb{Q}_p$ for some Prime $p$, and using properties of zeros of Power series over $\Bbb{Q}_p$ (Strassman's Theorem).

See also Arithmetic Progression, p-adic Number, Strassman's Theorem