Combinatorial properties of lazy expansions in Cantor real bases

Abstract

The lazy algorithm for a real base β is generalized to the setting of Cantor bases β = (βn) 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 a0a1a2 • • • such that each letter an belongs to {0, . . . , ⌈βn⌉ − 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.

Spectrum, algebraicity and normalization in alternate bases

“…Alternate bases are particular cases of Cantor real bases, which were introduced by the first two authors in [6] and then studied in [7,8]. A Cantor real base is a sequence β = (β n ) n∈N of real numbers greater than 1 such that +∞ n=0 β n = +∞.…”

“…In [6], the classical combinatorial results of the greedy β-expansions were generalized to the framework of Cantor re al bases. In [8], the lazy combinatorial properties of Cantor real bases were investigated. Moreover, in [7], the classical dynamical results of greedy and lazy β-expansions were generalized while focusing on periodic Cantor real bases β = (β 0 , .…”