ducci sequences using p-adic evaluation considering a starting sequence of integer $(a_1,a_2,...,a_n) \in \mathbb{Z^n}$ let it apply the ducci operator $D$
that act like this
\begin{equation}
\mathbb{Z^n}\rightarrow \mathbb{Z^n} \\
 (a_1,a_2,...,a_n)\mapsto (|a_1-a_2|,|a_2-a_3|,...,|a_n-a_1|)
\end{equation}
Let $D^k$ the Ducci operator acting $k$ times.
It is know that if $n=2^j$ then $D^k$ converge to the zero sequence $(0,0,...,0)$, in a finite number of steps. Otherwise it will converge to zero ot to a periodic sequence.
I was wondering if there is any results in the p-adic integers. Let rephase:
$(a_1,a_2,...,a_n) \in \mathbb{Z_p^n}$ where $\mathbb{Z_p}$ is the p-adic integers ring. Let $D_p$ the p-adic ducci operator acting
\begin{equation}
\mathbb{Z_p^n}\rightarrow \mathbb{Z_p^n} \\
 (a_1,a_2,...,a_n)\mapsto (|a_1-a_2|_p,|a_2-a_3|_p,...,|a_n-a_1|_p)
\end{equation}
Where $|x|_p$ is the p-adic valuation. Is is obvius that if $|a_i-a_{i+1}|_p$ is in the set $0,1,p^2,..p^{p-1}$ but I cannot go further, in deciding if applying the p-adic Ducci operator will create a sequence in $\mathbb{Z^p}$ that converges or becomes periodic. I google a while but didn't find any meaningfull reply.
Does someone faced this problem?
Thanks in advance
Paolo
 A: Things go very smoothly once we formalize the setting. First of all, fix a positive (rational) integer $n$, define the set $P = \{0\} \cup \{p^k : k \in \Bbb{Z}\}$, and consider the map
$$
\begin{align}
&D_p \colon &\Bbb{Q}_p^n &\to P^n \subset \Bbb{Q}_p^n\\
&&(a_1,\dotsc,a_n) &\mapsto (|a_1 - a_2|_p, |a_2 - a_3|_p,\dotsc, |a_n - a_1|_p).
\end{align}
$$
Then, given a tuple $\alpha = \alpha^{(0)} \in \Bbb{Q}_p^n$, consider the $p$-adic Ducci sequence $\{\alpha^{(k)}\}_{k = 0}^{\infty}$ with $\alpha^{(k)} = D_p^k(\alpha)$ for every $k \geq 1$. Now observe that, up to dropping the first term of the sequence, we may assume that $\alpha \in P^n$. Furthermore, by the ultrametric inequality we know that for every $x,y \in P$ we have
$$
\begin{cases}
|x - y|_p = \max(|x|_p,|y|_p) & \text{if } x \neq y \\
0 & \text{if } x = y
\end{cases}
$$
thus $p^{-\nu} \leq a_i^{(k)} \leq p^{\nu}$ for every $i \in \{1,\dotsc,n\}$ and $k \geq 0$, where $\nu = \max|v_p(a_i^{(0)})|$. In particular, this means that there are only finitely many possible values for $\alpha^{(k)}$, so by the pidgeonhole principle there must be some $r,s > 0$ such that $\alpha^{(r)} = \alpha^{(r+s)}$. But then
$$
\alpha^{(r + i + hs)} = \alpha^{(r+i)} \quad \text{for every } 0 \leq i < s \text{ and } h \geq 0
$$
i.e. $\{\alpha^{(k)}\}_{k = 0}^{\infty}$ is ultimately periodic.
We can actually go much further than that: with little work we can adapt the proof of lemma 3 in [2] to $D_p$, so the study of the periods of $p$-adic Ducci sequences reduces to that of the periods of Ducci sequences in $\Bbb{F}_2^n$. In particular, the proofs in section 3 of [1] work in this case without modification. For example, this means that if $n$ is a power of $2$, then for every $p$-adic Ducci sequence $\{\alpha^{(k)}\}$ there is a $K \geq 0$ such that$\alpha^{(k)} = 0$ for all $k \geq K$.

[1] Ehrlich, Amos. Periods in Ducci’s n-number game of differences. The Fibonacci Quart 28 (1990): 302-305.
[2] Furno, Anne Ludington. Cycles of differences of integers. Journal of Number Theory 13.2 (1981): 255-261.
