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Note establishes equivalence of algebraic and linear independence when Liouville bases are the same, suggesting deeper connections.
In 1985, Adams [‘The algebraic independence of certain Liouville continued fractions’, Proc. Amer. Math. Soc. 4 (1985), 512–516] proved that any finite number of Liouville series in different bases are algebraically independent. In this paper, we prove the equivalence between algebraic and linear independence when the bases are equal.
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Luiz Gustavo Dalpizol (2026) studied this question.
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