On the conjecture of Erdős, Joò and Komornik for p-adic numbers

The aim of this study is to give a partial answer to the analogue conjecture of Erdős, Joò and Komornik in the field of p-adic numbers. We prove that if \(\beta \) $\beta$ is a Pisot-Chabauty p-adic number, then the quantity \(l^m(\beta )\) $l^m(\beta )$ is strictly positive where \(\begin{aligned} l^m(\beta )= \inf \{|x|_p: x\in \Lambda _m(\beta )-\Lambda _m(\beta ), x\ne 0\} \end{aligned}\) $$\begin{array}{c}\hfill l^m(\beta )={inf\{|x|}_p:x\in {\mathrm{\Lambda }}_m(\beta )-{\mathrm{\Lambda }}_m(\beta ),x\ne 0\}\end{array}$$ and \(\begin{aligned} \Lambda _m(\beta )=\{\sum _{i=0}^n a_i \beta ^i \ : a_i\in \mathbb {Z}[\frac{1}{p}]\cap [0,1[ \ , \ |a_i|_p\le p^m \}. \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Lambda }}_m(\beta )=\{\sum _{i=0}^n{a}_i{\beta }^i:a_i\in \mathbb{Z}[\frac{1}{p}]\cap [0,1[,|a_i{|}_p\le p^m\}.\end{array}$$