×

zbMATH — the first resource for mathematics

Pisot numbers, primitive matrices and beta-conjugates. (Nombres de Pisots, matrices primitives et bêta-conjugués.) (French. English summary) Zbl 1285.11131
A subset \(\mathbb H\) of \(\mathbb N\) is relatively dense if there exists an integer \(m\) such that, for all \(n\in\mathbb N\), one at least of the numbers \(n,n+1,\dots,n+m\) belongs to \(\mathbb H\). A square matrix with all coefficient \(\geq 0\) is said positive. A positive matrix is said primitive if there exists an integer \(k\) such that all the coefficients of \(B^k\) are \(>0\). Perron has proved that a primitive matrix has a positive eigenvalue strictly greater than the module of all its other eigenvalues (called the dominant eigenvalue of the matrix).
A Perron number is an algebraic integer strictly greater than the module of all its conjugates. D. A. Lind [Ergodic Theory Dyn. Syst. 4, 283–300 (1984; Zbl 0546.58035)] has proved that: Every Perron number is the dominant eigenvalue of a matrix \(B\) with coefficients in \(\mathbb N\). A Pisot number is a real algebraic integer \(>1\) whose all conjugates are of module \(<1\). The author proves:
Let \(\beta\) be Pisot number of degree \(d\). There exists an integer \(n_0\) such that for every \(n>n_0\), we can find a primitive square matrix \(B_n\) of order \(d\), with coefficients in \(\mathbb N\), whose \(\beta^n\) is an eigenvalue.
The only eigenvalues of \(B_n\) are then \(\beta^n\) and its conjugates and they are simple.
If the irreducible polynomial of \(\beta^n\) has coefficients \(\geq 0\), then we can chose for \(B_n\) its companion matrix. If not, we can find a matrix of form \[ \left( \begin{matrix} 1 & 0 & 0 & \dots &\cdot & 0& c_d\\ 1 & 0 & \dots & \dots&\cdot &0 & b_{d-1}\\ 0 & 1 & \dots& \dots &\cdot&\cdot \cdot& b_{d-2}\\ \cdot & 0 & 1 & \dots &\cdot& \cdot&\cdot\\ \cdot & \cdot & \dots & \dots&\cdot &\cdot & \cdot\\ 0 & \cdot & \cdot & \dots&\cdot &\cdot& b_2\\ 0 & 0 & 0 & \dots &0 & 1 &b_1 \end{matrix} \right) \] Now, let \(\beta>1\) be a real number \(>1\) and \(\beta=a_1/\beta+a_2/\beta^2+\dots+a_n/\beta^n+\dots\) be the \(\beta\)-expansion of \(\beta\). If the sequence \((a_n)_{n\geq 1}\) is ultimately periodic of period \(k\) after the rank \(n_0\) ( i.e. for any \(n\geq n_0\), we have \(a_{n+k}=a_n\)), then \(\beta\) is called a Parry number; \(\beta\) is then an algebraic integer, strictly dominant root of the Parry polynomial, \[ X^{n_0+k}-(a_1X^{n_0+k-1}+\dots+a_{n_0+k})-(X^{n_0}-(a_1X^{n_0}+\dots+a_0)), \] where \(n_0\) and \(k\) are supposed minimum. There are two cases: either the degree of \(\beta\) is equal to the degree of the Parry polynomial which is then the minimal polynomial of \(\beta\) with \(d=n_0+k\), or it is strictly smaller and the Parry polynomial has other roots than the conjugates of \(\beta\). These other roots are called \(\beta\)-conjugates or pirat conjugates of \(\beta\). It is known that Pisot numbers are Parry numbers (see [A. Bertrand, C. R. Acad. Sci., Paris, Sér. A 285, 419–421 (1977; Zbl 0362.10040); ibid. 289, 1–4 (1979; Zbl 0418.10049)]).
The author proves: Let \(\beta\) be a Pisot number of degree \(d\). Let \(\mathcal F_\beta\) be the set of integers \(n\) such that \(\beta^n\) have no \(\beta\)-conjugate (i.e. the minimal polynomial of \(\beta^n\) and its Parry polynomial are the same). Then \(\mathcal F_\beta\) is relatively dense in \([1,\infty[\).

MSC:
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11B85 Automata sequences
15A18 Eigenvalues, singular values, and eigenvectors
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J.P. Allouche and S. Shallit, Automatic sequences. Cambridge University Press, 2003. · Zbl 1086.11015
[2] M.J. Bertin, A. Decomps, M. Grandet, M. Pathiaux and J.P. Schreiber., Pisot and Salem numbers. Birkhauser, Bale, 1992. · Zbl 0772.11041
[3] A. Bertrand, Développements en bases de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris Sér. A-B, 285, No 6 (1977), A419-A421. · Zbl 0362.10040
[4] A. Bertrand, Répartition modulo un et développements en bases de Pisot. C. R. Acad. Sci. Paris Sér. A, 289, No 1 (1979), 1-4. · Zbl 0418.10049
[5] C. Frougny and J. Sakarovitch, Automatic conversion from Fibonacci representation in base phi, and a generalization. Internat. J. Algebra Comput., 9 (1999), 351-384. · Zbl 1040.68061
[6] F.R. Gantmacher, The Theory of Matrices. Vol 2, Chelsea, New-York, 1959. · Zbl 0927.15002
[7] G.H. Hardy and E.M. Wright, An introduction to the theory of numbers. Clarendon Press, Oxford, 1938. · JFM 64.0093.03
[8] D. Lind, The entropies of topological Markov shifts and a related class of algebraic integers. Ergodic Theory and Dynamical Systems 4 (2) (1984), 283-300. · Zbl 0546.58035
[9] W. Parry, On the \(β -\) expansions of real numbers. Acta Math. Acad. Sci. Hung., 11 (1960), 401-416. · Zbl 0099.28103
[10] V. Prasolov, Problèmes et Théorèmes d’Algèbre Linéaire. Cassini, Paris, 2008.
[11] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Sci. Hungar., 8 (1957), 477-493. · Zbl 0079.08901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.