Leigh.Samphier/Sandbox/Definition:Eventually Periodic P-adic Expansion

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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.

Sources