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 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} Can one show that these sets of numbers agree?
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2$\begingroup$ They definitely don't agree. The "original" Liouville number $\sum_{k=1}^\infty 10^{-k!}$ satisfies the first definition but not the second definition. $\endgroup$– Greg MartinMar 24, 2018 at 5:51
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$\begingroup$ Thanks for your answer...yet both sets are of measure 0 and dense Gδ...and surely the second set is contained in the first...but why do Avron and Simon coin them Liouville numbers, then? Only a misuse of terminology? $\endgroup$– Eduard TetzlaffMar 24, 2018 at 16:23
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$\begingroup$ Without being provided even a link to the paper, we can't even speculate. $\endgroup$– Greg MartinMar 24, 2018 at 16:57
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$\begingroup$ Indeed, my bad...this one: math.caltech.edu/SimonPapers/149.pdf The definition of Liouville numbers appears on p.5 and the result is Corollary 7.2. As far as I understand this definition is needed to prove Gordon's Theorem. $\endgroup$– Eduard TetzlaffMar 24, 2018 at 17:01
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