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Real Mahler Series

Let $α$ and $β$ be positive real numbers. Let $F(x) \in K[[x^Γ]]$ be a Hahn series. We prove that if $F(x)$ is both $α$-Mahler and $β$-Mahler then it must be a rational function, $F(x) \in K(x)$, assuming some non-degeneracy conditions on $α$ and $β$.