Pisot number
A Pisot number or Pisot – Vijayaraghavan number , named after Charles Pisot (1910–1984) and Tirukkannapuram Vijayaraghavan (1902–1955), is an integer algebraic number for which its conjugate , ..., without itself (i.e. the other roots of the minimal polynomial of ) all lie inside the unit circle: . With “=” instead of “<”, you get the definition of a Salem number , named after Raphaël Salem . Traditionally, the set of Pisot numbers is denoted by S and the set of Salem numbers by T.
properties
The powers of a pisot number are exponentially close to whole numbers:
Adriano M. Garsia proved in 1962 that the set of real numbers with = 0, 1, 2, ... and is discrete . It is an unsolved problem whether someone that is not a pisot number can have this property .
In 1944, Raphaël Salem used Fourier analytical methods to show that the set of Pisot numbers is a closed subset of the real numbers.
Examples
Any integer greater than 1 is a pisot number. Other examples of Pisot numbers are the positive solutions to the algebraic equations
for = 2, 3,…, a sequence with . In particular is the golden number
- = 1.61803 39887 49894 84820 ...
a pisot number. It is also the smallest accumulation point in the set of Pisot numbers (Dufresnoy and Pisot 1955). The two smallest pisot numbers are
- = 1.32471 79572 44746 02596 ...,
the real solution of , and
- = 1.38027 75690 97614 11567 ...,
the positive real solution of .
Applications
Applications of pisot numbers can be found in geometric measurement theory, in connection with Bernoulli convolutions , in dimension theory and graph theory in the construction of pisot graphs .
literature
- Charles Pisot : La répartition modulo 1 et les nombres algébriques . In: Annali della Scuola Normale Superiore di Pisa - Classe di Scienze , 7, 1938, pp. 205–248 (dissertation; French)
-
T. Vijayaraghavan : On the fractional parts of the powers of a number (English)
- I . In: Journal of the London Mathematical Society , 15, 1940, pp. 159-160
- II . In: Mathematical Proceedings of the Cambridge Philosophical Society , 37, 1941, pp. 349-357
- III . In: Journal of the London Mathematical Society , 17, 1942, pp. 137-138
- IV . In: Journal of the Indian Mathematical Society , 12, 1948, pp. 33-39
- Raphaël Salem : A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan . In: Duke Mathematical Journal , 11, 1944, pp. 103-107 (English)
- Jacques Dufresnoy, Charles Pisot: Étude de certaines fonctions méromorphes bornées sur le cercle unité. Application à un ensemble fermé d'entiers algébriques . In: Annales scientifiques de l'École Normale Supérieure , 72, 1955, pp. 69–92 (French)
- Raphaël Salem: Algebraic numbers and Fourier Analysis . Heath, Boston 1963 (English)
- Adriano M. Garsia: Arithmetic properties of Bernoulli convolutions . In: Transactions of the AMS , 102, 1962, pp. 409-432 (English)
- Adriano M. Garsia: Entropy and singularity of infinite convolutions . In: Pacific Journal of Mathematics , 13, 1963, pp. 1159–1169 (English)
- Yves Meyer : Algebraic numbers and harmonic analysis . North-Holland, Amsterdam 1972 (English)
- Marie-José Bertin, Annette Decomps-Guilloux, Marthe Grandet-Hugot, Martine Pathiaux-Delefosse, Jean-Pierre Schreiber: Pisot and Salem numbers . Birkhäuser, Basel 1992, ISBN 3-7643-2648-4 (English)
- James McKee, Chris Smith: Salem Numbers, Pisot Numbers, Mahler Measure, and Graphs . (PDF; 875 kB) In: Experimental Mathematics , 14, 2005, pp. 211–229 (English)
Web links
- David Terr: Pisot Number . In: MathWorld (English).
- David Boyd: Pisot number . In: Encyclopaedia of Mathematics , Springer, 2001 (English)
- Andrew Potter: Pisot numbers ( Memento from September 27, 2006 in the Internet Archive ; PDF) - simple introduction (English)