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The number $$0.10111001001000000000001$$ has continued fraction $$[0, 9, 1, 8, 9, 5, 1, 1, 5, 3, 1, 3, 1, 1, 4, 6, 1, 1, 8, 2, 5, 8, 1, 9, 9, 5, 2 , 8, 1, 1, 6, 4, 1, 1, 3, 1, 3, 5, 1, 1, 5, 9, 8, 1, 9]$$

So, the maximum values is $9$. But we are only at $23$ digits. Can we produce larger decimal expansions with the required property ? Perhaps arbitary large ones ?

Is there an irrational number, such that the decimal expansion contains only ones and zeros and the continued fraction contains no entry larger than $9$ ?

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    (+1) An interesting question on the intersection of Cantor-like sets. – 2017-01-19
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    $$0.1011010000001000000010000000000000$$ $$0000000010000000000000000000000000000000000000 0000001$$ is far better with $87$ digits! Seems that we can continute forever, if we add a digit $1$ and then insert enough $0$'s between the last two ones ... – 2017-01-19
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    But I would prefer sequences without such long $0$-sequences, if possible :) – 2017-01-19
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    A shorter example without long sequences is $$0.1011010011100100111101101110101$$ – 2017-01-19

0 Answers 0