\begin{definition}[Definition:Basis Expansion/Recurrence]
Let $b \in \N: b \ge 2$.
Let $x$ be a real number.
Let the basis expansion of $x$ in base $b$ be:
:$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$
Let there be a finite sequence of $p$ digits of $x$:
:$\tuple {d_{r + 1} d_{r + 1} \ldots d_{r + p} }$
such that for all $k \in \Z_{\ge 0}$ and for all $j \in \set {1, 2, \ldots, p}$:
:$d_{r + j + k p} = d_{r + j}$
where $p$ is the smallest $p$ to have this property.
That is, let $x$ be of the form:
:$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots d_r d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} d_{r + 2} \ldots d_{r + p} d_{r + 1} \ldots}_b$
That is, $\tuple {d_{r + 1} d_{r + 2} \ldots d_{r + p} }$ repeats from then on, or '''recurs'''.
Then $x$ is said to '''recur'''.
\end{definition}