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Real and \(p\)-adic expansions involving symmetric patterns. (English) Zbl 1113.11041

Inspired by a conjecture of Mahler and of Mendès France, the authors state the following problem: let \((a_n)_{n\geq 1}\) be a sequence of integers with values in \([0,p-1]\). Is it true that the real number \(\sum a_n p^{-n}\) and the \(p\)-adic number \(\sum a_n p^n\) are both algebraic if and only if they are both rational? While this question remains open, they prove that the answer is positive for the sequence \((a_n)_{n\geq 1}\) containing infinitely many palindromes “not too far’ from its beginning (their statement is of course quantitative, and more general).
Please note that reference [2] appeared [B. Adamczewski and Y. Bugeaud, Ann. Math. (2) 165, 547–565 (2007; Zbl 1195.11094)].

MSC:

11J81 Transcendence (general theory)
68R15 Combinatorics on words

Citations:

Zbl 1195.11094
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