Definition:Eventually 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 $\ldots d_n \ldots d_2 d_1 d_0 . d_{-1} d_{-2} \ldots d_{-m}$ be the canonical expansion of $x$.

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

$\tuple {d_{r + k - 1} \ldots d_{r+1} d_r }$

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

$d_{r + j + n k} = d_{r + j}$

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

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

$\ldots d_{r + k - 1} \ldots d_{r+1} d_r d_{r + k - 1} \ldots d_{r+1} d_r d_{r + k - 1} \ldots d_{r+1} d_r \ldots d_2 d_1 d_0 . d_{-1} d_{-2} \ldots d_{-m}$

That is, $\tuple {d_{r + k - 1} \ldots d_{r+1} d_r }$ repeats.

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

Also see

• Definition:Periodic P-adic Expansion

• Canonical P-adic Expansion of Rational is Eventually Periodic

Sources

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