Let be a number greater than 1, a Positive number, and

(1) |

Pisot (1938) proved that if is such that there exists a such that the series
converges, then is an Algebraic Integer whose conjugates all (except for itself)
have modulus , and is an algebraic Integer of the Field . Vijayaraghavan (1940) proved
that the set of Pisot-Vijayaraghavan numbers has infinitely many limit points. Salem (1944) proved that the set of
Pisot-Vijayaraghavan constants is closed. The proof of this theorem is based on the Lemma that for a
Pisot-Vijayaraghavan constant , there always exists a number such that
and the
following inequality is satisfied,

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

number | order | Polynomial | |

0 | 1.3247179572 | 3 | 1 0 |

1 | 1.3802775691 | 4 | 1 0 0 |

1.6216584885 | 16 | 1 2 2 1 0 0 1 2 2 1 | |

1.8374664495 | 20 | 1 0 1 0 1 0 1 0 0 1 0 1 0 1 |

All the points in less than are known (Dufresnoy and Pisot 1955). Each point of is a limit point from both sides of the set of Salem Constants (Salem 1945).

**References**

Boyd, D. W. ``Small Salem Numbers.'' *Duke Math. J.* **44**, 315-328, 1977.

Dufresnoy, J. and Pisot, C. ``Étude de certaines fonctions méromorphes bornées sur le cercle unité,
application à un ensemble fermé d'entiers algébriques.'' *Ann. Sci. École Norm. Sup.* **72**, 69-92, 1955.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, pp. 38 and 148, 1983.

Koksma, J. F. ``Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins.'' *Comp. Math.* **2**, 250-258, 1935.

Pisot, C. ``La répartition modulo 1 et les nombres algébriques.'' *Annali di Pisa* **7**, 205-248, 1938.

Salem, R. ``Sets of Uniqueness and Sets of Multiplicity.'' *Trans. Amer. Math. Soc.* **54**, 218-228, 1943.

Salem, R. ``A Remarkable Class of Algebraic Numbers. Proof of a Conjecture of Vijayaraghavan.'' *Duke
Math. J.* **11**, 103-108, 1944.

Salem, R. ``Power Series with Integral Coefficients.'' *Duke Math. J.* **12**, 153-172, 1945.

Siegel, C. L. ``Algebraic Numbers whose Conjugates Lie in the Unit Circle.'' *Duke Math. J.* **11**, 597-602, 1944.

Vijayaraghavan, T. ``On the Fractional Parts of the Powers of a Number, II.'' *Proc. Cambridge Phil. Soc.* **37**, 349-357, 1941.

© 1996-9

1999-05-25