Pisot Number
A Pisot number is a positive algebraic integer greater than 1 all of whose conjugate elements have absolute value less than 1. A real quadratic algebraic integer greater than 1 and of degree 2 or 3 is a Pisot number if its norm is equal to 
. The golden ratio 
(denoted 
when considered as a Pisot number) is an example of a Pisot number since it has degree two and norm 
.
The smallest Pisot number is given by the positive root 
(OEIS A060006) of
 |
(1)
|
known as the plastic constant. This number was identified as the smallest known by Salem (1944), and proved to be the smallest possible by Siegel (1944).
Pisot constants give rise to almost integers. For example, the larger the power to which 
is taken, the closer 
, where 
is the floor function, is to either 0 or 1 (Trott 2004). For example, the spectacular example 
is within 
of an integer (Trott 2004, pp. 8-9).
The powers of 
for which this quantity is closer to 0 are 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 17, ... (OEIS A051016), while those for which it is closer to 1 are 2, 9, 10, 13, 15, 16, 18, 20, 21, 23, ... (OEIS A051017).
Siegel also identified the second smallest Pisot numbers as the positive root 
(OEIS A086106) of
 |
(2)
|
showed that 
and 
are isolated, and showed that the positive roots of each polynomial
 |
(3)
|
for 
, 2, 3, ...,
 |
(4)
|
for 
, 5, 7, ..., and
 |
(5)
|
for 
, 5, 7, ... are Pisot numbers.
All the Pisot numbers less than 
are known (Dufresnoy and Pisot 1955). Some small Pisot numbers and their polynomials are given in the following table. The latter two entries are from Boyd (1977).
| number |
Sloane |
order |
polynomial coefficients |
| 1.3247179572 |
A060006 |
3 |
1 0   |
| 1.3802775691 |
A086106 |
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  |
Pisot numbers originally arose in the consideration of
 |
(6)
|
where 
denotes the fractional part of 
and 
is the floor function. Letting 
be a number greater than 1 and 
a positive number, for a given 
, the sequence of numbers 
for 
, 2, ... is an equidistributed sequence in the interval (0, 1) when 
does not belong to a 
-dependent exceptional set 
of measure zero (Koksma 1935). Pisot (1938) and Vijayaraghavan (1941) independently studied the exceptional values of 
, and Salem (1943) proposed calling such values Pisot-Vijayaraghavan numbers.
Pisot (1938) subsequently proved the fact that if 
is chosen such that there exists a 
for which the series
 |
(7)
|
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 numbers has infinitely many limit points. Salem (1944) proved that the set of Pisot numbers is closed. The proof of this theorem is based on the lemma that for a Pisot number 
, there always exists a number 
such that 
and the following inequality is satisfied:
 |
(8)
|
REFERENCES:
Bell, J. P. and Hare, K. G. "Properties of 
for 
a Pisot number." http://www.math.uwaterloo.ca/~kghare/Preprints/PDF/P17_Zq.pdf.
Bertin, M. J. and Pathiaux-Delefosse, A. Conjecture de Lehmer et petits nombres de Salem. Kingston: Queen's Papers in Pure and Applied Mathematics, 1989.
Bertin, M. J.; Decomps-Guilloux, A.; Grandet-Hugot, M.; Pathiaux-Delefosse, M.; and Schreiber, J. P. Pisot and Salem Numbers. Basel: Birkhäuser, 1992.
Borwein, P. and Hare, K. G. "Some Computations on Pisot and Salem Numbers." CECM-00:148, 18 May. http://www.cecm.sfu.ca/preprints/2000pp.html#00:148.
Boyd, D. W. "Small Salem Numbers." Duke Math. J. 44, 315-328, 1977.
Boyd, D. W. "Pisot and Salem Numbers in Intervals of the Real Line." Math. Comput. 32, 1244-1260, 1978.
Boyd, D. W. "Pisot Numbers in the Neighbourhood of a Limit Point. II." Math. Comput. 43, 593-602, 1984.
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Dubickas, A. "A Note on Powers of Pisot Numbers." Publ. Math. Debrecen 56, 141-144, 2000.
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.
Erdős, P.; Joó, M.; and Schnitzer, F. J. "On Pisot Numbers." Ann. Univ. Sci. Budapest, Eőtvős Sect. Math. 39, 95-99, 1997.
Katai, I. and Kovacs, B. "Multiplicative Functions with Nearly Integer Values." Acta Sci. Math. 48, 221-225, 1985.
Koksma, J. F. "Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins." Comp. Math. 2, 250-258, 1935.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 38 and 148, 1983.
Luca, F. "On a Question of G. Kuba." Arch. Math. (Basel) 74, 269-275, 2000.
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.
Sloane, N. J. A. Sequences A051016, A051017, A060006, and A086106 in "The On-Line Encyclopedia of Integer Sequences."
Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.
Vijayaraghavan, T. "On the Fractional Parts of the Powers of a Number, II." Proc. Cambridge Phil. Soc. 37, 349-357, 1941.
