Mahler's theorem

Remarks

The binomial polynomials $\binom{x}{k}$ take integer values on $\N$, which is dense in $\Z_p$, so they map $\Z_p$ into $\Z_p$; hence $\abs{\binom{x}{k}}_p \le 1$ and the series converges uniformly when $a_k \to 0$. On $\N$ the expansion is Newton's forward difference formula, and both sides are continuous, so the main point is that $a_k \to 0$. This is proved by showing that $\Delta^{p^m}$ is small on continuous functions for large $m$, using the expansion of $\Delta^{p^m}$ in shifts and the divisibility of binomial coefficients by $p$. Moreover, $\sup_{x \in \Z_p} \abs{f(x)}_p = \max_k \abs{a_k}_p$, so the functions $\binom{x}{k}$ form an orthonormal basis of the Banach space of continuous functions $\Z_p \to \Q_p$. Example: for $u \in \Z_p$ with $\abs{u - 1}_p < 1$, the function $x \mapsto u^x$ is continuous, and its expansion is $\sum_k (u - 1)^k \binom{x}{k}$, with $a_k = (u-1)^k \to 0$.

Going deeper

For a student

The theorem identifies the space of continuous functions $\Z_p \to \Q_p$ with the space of null sequences, isometrically for the sup norm. It is a basic tool in $p$-adic analysis, for example to construct $p$-adic interpolations of functions defined on the natural numbers, and to describe $p$-adic measures by their moments. It is the $p$-adic counterpart of the Newton series expansion, where in the real case convergence is a delicate question.