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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?

Eduard Tetzlaff
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1 Answers1

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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.

zhoraster
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  • Thanks for your answer...yet both sets are of measure 0 and dense $G_\delta$... Why do Avron and Simon coin them Liouville numbers, then? – Eduard Tetzlaff Mar 24 '18 at 15:28
  • As I said, seems to be some old misuse of terminology. I was not able to trace it to the origin, which may well be Avron and Simon's paper. – zhoraster Mar 24 '18 at 15:52
  • Ok thanks! I also couldnt find any other source for that (and obviously the monograph by Cycon, Froese, Kirsch and Simon). They need the second bound, though, to apply Gordon's theorem... – Eduard Tetzlaff Mar 24 '18 at 16:21