Combinatorial properties of lazy expansions in Cantor real bases
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The lazy algorithm for a real base β is generalized to the setting of Cantor bases β=(β_n)_n∈ℕ introduced recently by Charlier and the author. To do so, let x_β be the greatest real number that has a β-representation a_0a_1a_2⋯ such that each letter a_n belongs to {0,…,⌈β_n ⌉ -1}. This paper is concerned with the combinatorial properties of the lazy β-expansions, which are defined when x_β<+∞. As an illustration, Cantor bases following the Thue-Morse sequence are studied and a formula giving their corresponding value of x_β is proved. First, it is shown that the lazy β-expansions are obtained by "flipping" the digits of the greedy β-expansions. Next, a Parry-like criterion characterizing the sequences of non-negative integers that are the lazy β-expansions of some real number in (x_β-1,x_β] is proved. Moreover, the lazy β-shift is studied and in the particular case of alternate bases, that is the periodic Cantor bases, an analogue of Bertrand-Mathis' theorem in the lazy framework is proved: the lazy β-shift is sofic if and only if all quasi-lazy β^(i)-expansions of x_β^(i)-1 are ultimately periodic, where β^(i) is the i-th shift of the alternate base β.
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