Different characterizations of Liouville numbers

Question

Usually, Liouville numbers are defined as follows: $x$ is Liouville if for ever $i\in\mathbb N$ there exist $n,m\in\mathbb Z$ such that \begin{equation} \left|x-\frac nm\right|<\frac1{m^i}. \end{equation} In their 1982 paper on almost-periodic Schrödinger operators, however, Avron and Simon use the following definition: $x$ is Liouville if for ever $i\in\mathbb N$ there exist $n,m\in\mathbb Z$ such that \begin{equation} \left|x-\frac nm\right|<\frac1{i^m}. \end{equation} Do these sets of numbers agree? If yes, how can one show that?

Answer 1

This seems to be a long-time misuse of terminology. The second definition is not of a Liouville number, it is much stronger. Liouville number is a number which can be approximated by rationals at any power rate. The second definition assumes that a number can be approximated at any exponential rate.