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Abstract

Let $Q=(b_n)_{n\geq {1}}$ be a sequence of positive integers with ${b_n}\geq {2}$ for all integers $n$. Let $\alpha $ be a non-zero real number written in $Q$-ary expansion. In 2007, Adamczewski and Bugeaud, using the subspace theorem, proved, under some conditions, that given two real numbers written in $b$-ary expansion either they are equivalent (i.e., their $b$-expansions have the same tail) or one of them is a transcendental number. In this article, we prove analogous results for real numbers written in $Q$-ary expansion.