On the conjecture of Erdős, Joò and Komornik for \(p\)-adic numbers (Q6590166)

The authors first define a Pisot-Chabauty number (PC number) as a \(p\)-adic analogue of a Pisot number in \({\mathbb Q}_p\). Their main result (Theorem 4.3) asserts that if \(\beta \in {\mathbb Q}_p\) is a PC number, then \(l^m(\beta)>0\) for each positive integer \(m\). Here, \(l^m(x)\) is defined as \(\inf |x|_p\), where \(x \ne 0\) is of the form \(\Lambda_m(\beta)-\Lambda_m(\beta)\) and \(\Lambda_m(\beta)\) is the set of numbers \(\sum_{i=0}^n a_i \beta^i\), where \(a_i \in {\mathbb Z}[1/p] \cap [0,1]\) satisfy \(|a_i|_p \leq p^m\).