SETS OF RANGE UNIQUENESS IN $p$-ADIC FIELDS

Abstract

We study sets of range uniqueness (SRUs) for analytic functions inside a disc of an algebraically closed field $K$ complete with respect to an ultrametric absolute value. The SRUs we obtain are converging sequences. We first obtain results that look like those known in $\mathbb{C}$ but involve a weaker hypothesis than in $\mathbb{C}$: let $(a_n)$ be a sequence of limit $a$ in a disc $d(a,r^-)$ such that $|a_n-a|$ is a strictly decreasing sequence. If the sequence $(a_n)$ does not make an SRU for the set $\mathcal{A}(d(a,r^-))$ of analytic functions inside $d(a,r^-)$, then, for a certain integer $k\in\mathbb{Z}$, the sequence

$$ \bigg(\frac{a_{n+k}-a}{a_n-a}\bigg) $$

has a finite limit in $K$ and the sequence

$$ \bigg(\frac{\log|a_{n+k}-a|}{\log|a_n-a|}\bigg) $$

has a finite rational limit. Next, we show that if the sequence

$$ \frac{\log(a_{n+1}-a)}{\log(a_n-a)} $$

converges to a limit $b\geq1$ in such a way that $-b\log|a_{n}-a|\lt-b\log|a_{n+1}-a|$ and if $\log|a_{n}-a|-b\log|a_{n+1}-a|$ has limit $0$ or $+\infty$ and if $b^k\notin\mathbb{Q}$ whenever $b>1$ and $k\in \mathbb{N}^*$, then the sequence $(a_n)$ is an SRU for $\mathcal{A}(d(a,r^-))$. In particular, for every $\gamma\in\;]0,1[\;\cup\;]1,+\infty[$, $L\in\mathbb{Q}\;\cap\;]0,+\infty[$ and $b\geq 1$, there exist SRUs for $\mathcal{A}(d(a,r^-))$ of the form $\{a_n\mid n\in\mathbb{N}\}$ such that

$$ \lim_{n\rightarrow+\infty}\frac{-\log|a_n-a|}{b^nn^\gamma}=L. $$

For example, if $\gamma\in\mathbb{N}$ with $\gamma\neq0,1$, there exist SRUs of the form $\{a_n\mid n\in\mathbb{N}\}$ such that $-\log |a_n-a|=Ln^\gamma$ for all $n\in\mathbb{N}^*$. The latter result ceases to hold when $\gamma=1$. Many examples and counterexamples are provided.