Expansions in Cantor real bases
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We introduce and study series expansions of real numbers with an arbitrary Cantor real base β=(β_n)_n∈ℕ, which we call β-representations. In doing so, we generalize both representations of real numbers in real bases and through Cantor series. We show fundamental properties of β-representations, each of which extends existing results on representations in a real base. In particular, we prove a generalization of Parry's theorem characterizing sequences of nonnegative integers that are the greedy β-representations of some real number in the interval [0,1). We pay special attention to periodic Cantor real bases, which we call alternate bases. In this case, we show that the β-shift is sofic if and only if all quasi-greedy β^(i)-expansions of 1 are ultimately periodic, where β^(i) is the i-th shift of the Cantor real base β.
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