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. ${\displaystyle \alpha >1}$ ${\displaystyle \alpha _{2}}$ ${\displaystyle \alpha _{d}}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \rho =\max\{|\alpha _{2}|,\ldots ,|\alpha _{d}|\}<1}$ ${\displaystyle \max\{|\alpha _{2}|,\ldots ,|\alpha _{d}|\}=1}$

properties

The powers of a pisot number are exponentially close to whole numbers: ${\displaystyle \alpha ^{k}}$ ${\displaystyle \alpha }$

${\displaystyle \min {\bigl \{}|\alpha ^{k}-z|\,{\big |}\,z\in \mathbb {Z} {\bigr \}}\leq (d-1)\rho ^{k}}$

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 . ${\displaystyle |\varepsilon _{n}\,\alpha ^{n}+\varepsilon _{n-1}\,\alpha ^{n-1}+\ldots +\varepsilon _{1}\,\alpha +\varepsilon _{0}|}$ ${\displaystyle n}$ ${\displaystyle \varepsilon _{0},\ldots ,\varepsilon _{n}\in \{-1,0,+1\}}$ ${\displaystyle \alpha >1}$

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 ${\displaystyle \beta _{n}}$

${\displaystyle x^{n}-x^{n-1}-x^{n-2}-\dots -1=0,}$

for = 2, 3,…, a sequence with . In particular is the golden number ${\displaystyle n}$ ${\displaystyle \beta _{n}\to 2}$

${\displaystyle \Phi =\beta _{2}}$ = 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

${\displaystyle \theta _{1}}$ = 1.32471 79572 44746 02596 ...,

the real solution of , and ${\displaystyle x^{3}-x-1=0}$

${\displaystyle \theta _{2}}$ = 1.38027 75690 97614 11567 ...,

the positive real solution of . ${\displaystyle x^{4}-x^{3}-1=0}$

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)

Individual evidence

1. ↑ Follow A001622 in OEIS
2. ↑ Follow A060006 in OEIS
3. ↑ Follow A086106 in OEIS
4. ↑ see also Michel Mendès-France: Book Review . In: Bulletin of the AMS , 29, 1993, pp. 274-278