---
tag: D115PG3T7K
kind: statement
role: theorem
name: Mahler's theorem
msc: [11S80, 46S10, 39A70]
uses: [GBN9F8FAQ3, CHT5JYM6QW, 0P27W9RW0G, E38801KVJ1, 08FPABDQ4C]
about: [Y2HXSEA2EE]
sources:
  - url: https://en.wikipedia.org/wiki/Mahler%27s_theorem
same_as:
  wikidata: Q3527118
  wikipedia: Mahler's theorem
---

## Statement

Let $p$ be a prime, let $\Z_p$ be the ring of [[$p$-adic integers|GBN9F8FAQ3]], and let $\Q_p$ be the field of [[$p$-adic numbers|CHT5JYM6QW]].
For $k \in \N$ and $x \in \Z_p$ let $\binom{x}{k} = \frac{x(x-1)\cdots(x-k+1)}{k!}$.
Let $f \colon \Z_p \to \Q_p$ be a [[continuous function|0P27W9RW0G]], and let $a_k = \Delta^k f(0)$ be its [[$k$-th forward difference|E38801KVJ1]] at $0$ with step $1$.
Then $a_k \to 0$ in $\Q_p$, and
$$f(x) = \sum_{k=0}^{\infty} a_k \binom{x}{k} \quad \text{for all } x \in \Z_p,$$
where the series [[converges uniformly|08FPABDQ4C]] on $\Z_p$.
Conversely, for every sequence $(a_k)$ in $\Q_p$ with $a_k \to 0$, this series defines a continuous function $f \colon \Z_p \to \Q_p$ with $\Delta^k f(0) = a_k$ for all $k$.

## 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|DDHCMKHD2T]], 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 $a_k = (u-1)^k \to 0$.

## Explanations

### L1 human

Every continuous function on the $p$-adic integers is given by a Newton series that converges everywhere.
The coefficients are the forward differences at $0$, which can be read off from the values $f(0), f(1), f(2), \dots$.
Over the real numbers nothing like this holds, since a continuous function on $\R$ is far from being fixed by its values at the natural numbers.

### L2 human

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|Y2HXSEA2EE]] expansion, where in the real case convergence is a delicate question.