Primitive substitutive numbers are closed under rational multiplication
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},
     mrnumber = {1828248},
     zbl = {0930.11008},
     language = {en},
     url = {http://archive.numdam.org/item/JTNB_1998__10_2_315_0/}
}
TY  - JOUR
AU  - Ketkar, Pallavi
AU  - Zamboni, Luca Q.
TI  - Primitive substitutive numbers are closed under rational multiplication
JO  - Journal de théorie des nombres de Bordeaux
PY  - 1998
SP  - 315
EP  - 320
VL  - 10
IS  - 2
PB  - Université Bordeaux I
UR  - http://archive.numdam.org/item/JTNB_1998__10_2_315_0/
LA  - en
ID  - JTNB_1998__10_2_315_0
ER  - 
%0 Journal Article
%A Ketkar, Pallavi
%A Zamboni, Luca Q.
%T Primitive substitutive numbers are closed under rational multiplication
%J Journal de théorie des nombres de Bordeaux
%D 1998
%P 315-320
%V 10
%N 2
%I Université Bordeaux I
%U http://archive.numdam.org/item/JTNB_1998__10_2_315_0/
%G en
%F JTNB_1998__10_2_315_0
Ketkar, Pallavi; Zamboni, Luca Q. Primitive substitutive numbers are closed under rational multiplication. Journal de théorie des nombres de Bordeaux, Tome 10 (1998) no. 2, pp. 315-320. http://archive.numdam.org/item/JTNB_1998__10_2_315_0/

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

[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 | Zbl

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

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

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

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

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

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

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

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