Definition:Periodic P-adic Expansion

Definition

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.

Let $x \in \Q_p$.

Let $\ldots d_n \ldots d_2 d_1 d_0$ be the canonical expansion of $x$.

Let there be a finite sequence of $k$ digits of $x$:

$\tuple {d_{k - 1} \ldots d_1 d_0 }$

such that for all $n \in \Z_{\ge 0}$ and for all $j \in \set {1, 2, \ldots, k}$:

$d_{j + n k} = d_j$

where $k$ is the smallest $k$ to have this property.

That is, let the canonical expansion of $x$ be of the form:

$\ldots d_{k - 1} \ldots d_1 d_0 d_{k - 1} \ldots d_1 d_0 d_{k - 1} \ldots d_1 d_0$

That is, $\tuple {d_{k - 1} \ldots d_1 d_0 }$ repeats.

Then the canonical expansion of $x$ is said to be periodic.

2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.6$ The $p$-adic expansion of rational numbers: Theorem $1.38$