This question might be too simple and I just don't see something very obvious, so I apologize in advance if that is so.
Let $p$ be a prime number and let $\mathbb Z_p$ denote the ring of $p$-adic integers. Let $\alpha \in \mathbb Z_p$ be an algebraic number of degree $d \geq 2$ (i.e. with a minimal polynomial in $\mathbb Z[x]$ of degree $d \geq 2$). This number can be written in the form
$$ \alpha = \sum\limits_{i = 0}^\infty a_ip^i, $$
where $a_i \in \{0, 1, \ldots, p - 1\}$. Intuitively it should be the case that, as $i$ gets bigger, we cannot have too many $a_i$'s in succession being zero. In other words, if for some sufficiently large $i_0$ it is the case that
$$a_{i_0} = a_{i_0+1} = \ldots = a_{i_0 + h} = 0,$$
then $h$ cannot be too big. Otherwise, the integer $m = \sum\limits_{i = 0}^{i_0 - 1}a_ip^i$ would approximate $\alpha$ "too well". Is there any way to quantify this heuristics and prove it, whether effectively or not? (optimally, I want to know an upper bound on $h$)
