Primitive substitutive numbers are closed under rational multiplication
Pallavi Ketkar ; Luca Q. Zamboni
Journal de Théorie des Nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 315-320.

Soit M(r) l’ensemble des réels α dont le développement en base r contient une queue qui est l’image d’un point fixe d’une substitution primitive par un morphisme de lettres. Nous démontrons que l’ensemble M(r) est stable par multiplication par les rationnels, mais non stable par addition.

Let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution. We show that the set M(r) is closed under multiplication by rational numbers, but not closed under addition.

@article{JTNB_1998__10_2_315_0,
     author = {Ketkar, Pallavi and Zamboni, Luca Q.},
     title = {Primitive substitutive numbers are closed under rational multiplication},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {315--320},
     publisher = {Universit\'e Bordeaux I},
     volume = {10},
     number = {2},
     year = {1998},
     doi = {10.5802/jtnb.231},
     zbl = {0930.11008},
     mrnumber = {1828248},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.231/}
}
Pallavi Ketkar; Luca Q. Zamboni. Primitive substitutive numbers are closed under rational multiplication. Journal de Théorie des Nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 315-320. doi : 10.5802/jtnb.231. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.231/

[AlMe] J.-P. Allouche, M. Mendès France, Quasicrystal ising chain and automata theory. J. Statist. Phys. 42 (1986), 809-821. | MR 833222 | Zbl 0641.10043

[AlZa] J.-P. Allouche, L.Q. Zamboni, Algebraic irrational binary numbers cannot be fixed points of non-trivial constant length or primitive morphisms. J. Number Theory 69 (1998), 119-124. | MR 1611101 | Zbl 0918.11016

[De] F.M. Dekking, Iteration of maps by an automaton. Discrete Math. 126 (1994), 81-86. | MR 1264477 | Zbl 0795.68158

[Du] F. Durand, A characterization of substitutive sequences using return words. Discrete Math. 179 (1998), 89-101. | MR 1489074 | Zbl 0895.68087

[FeMa] S. Ferenczi, C. Mauduit, Transcendence of numbers with a low complexity expansion. J. Number Theory 67 (1997), 146-161. | MR 1486494 | Zbl 0895.11029

[HoZa] C. Holton, L.Q. Zamboni, Iteration of maps by primitive substitutive sequences. (1998), to appear in Discrete Math. | MR 1796154

[Le] S. Lehr, Sums and rational multiples of q-automatic sequences are q-automatic. Theoret. Comp. Sys. 108 (1993), 385-391. | MR 1202029 | Zbl 0768.11013

[LoPo] J.H. Loxton, A. Van Der Poorten, Arithmetic properties of automata: regular sequences. J. Reine Angew. Math. 392 (1988), 57-69. | MR 965057 | Zbl 0656.10033

[Qu] M. Queffélec, Substitution Dynamical Systems-Spectral Analysis. Lecture Notes in Math. 1294, Springer-Verlag, Berlin- New York, 1987. | MR 924156 | Zbl 0642.28013

[RiZa] R. Risley, L.Q. Zamboni, A generalization of Sturmian flows; combinatorial structure and transcendence. preprint 1998.