Pisot number
Pisot–Vijayaraghavan number
A real algebraic integer , all of whose other Galois conjugates have absolute value strictly less than 1 (cf. also Galois theory). That is, \theta satisfies a polynomial equation of the form P ( x ) = x ^ { n } + a _ { 1 } x ^ { n - 1 } + \dots + a _ { n }, where the a_k are integers, a _ { n } \neq 0 and the roots of P ( x ) other than \theta all lie in the open unit circle | x | < 1. The set of these numbers is traditionally denoted by S. Every positive integer n > 1 is a Pisot number, but a more interesting example is the golden ratio ( 1 + \sqrt { 5 } ) / 2. Every real number field K contains infinitely many Pisot numbers of degree equal to [ K : \mathbf{Q} ], and, in fact, every real number field K can be generated by Pisot numbers, even by Pisot units (a _ { n } = 1).
The Pisot numbers have the interesting property that if 0 < \lambda \in \mathbf{Z} ( \theta ), then \| \lambda \theta ^ { n } \| \rightarrow 0 as n \rightarrow \infty, where here \| x \| = \operatorname { dist } ( x , \mathbf{Z} ) denotes the distance from x to the nearest integer. It is an open question whether this property characterizes S among the real numbers \theta > 1 (Pisot's conjecture). An important result of Ch. Pisot in this direction is that if \theta > 1 and \lambda > 0 are real numbers for which \sum _ { n = 0 } ^ { \infty } \| \lambda \theta ^ { n } \| ^ { 2 } < \infty, then \theta \in S and \lambda \in \mathbf{Q} ( \theta ) [a1].
The unusual behaviour of the powers of Pisot numbers leads to applications in harmonic analysis, [a3], [a5], dynamical systems theory (cf. also Dynamical system) [a6] and the theory of quasi-crystals [a4]. For example, if \theta > 1, then the set of powers \{ 1 , \theta , \theta ^ { 2 } , \ldots \} is harmonious if and only if \theta is a Pisot number or a Salem number [a3]. The Bragg spectrum of the diffraction pattern of a self-similar tiling (cf., e.g., Voronoi lattice types) is non-trivial if and only if the scaling factor of the tiling is a Pisot number [a4].
A surprising fact is that S is a closed and hence nowhere-dense subset of the real line [a5]. The derived sets S ^ { ( n ) } are all non-empty and \operatorname { min } S ^ { ( n ) } \rightarrow \infty as n \rightarrow \infty. (Here S ^ { \prime } = S ^ { ( 1 ) } denotes the set of limit points of S, S ^ { \prime \prime } = S ^ { ( 2 ) } the set of limit points of S ^ { \prime }, etc., cf. also Limit point of a set). The order type of S is described in [a2]. The smallest elements of S, S ^ { \prime } and S ^ { \prime \prime } are explicitly known [a1].
There is an intimate relationship between the set S of Pisot numbers and the set T of Salem numbers. It is known that S \subset T ^ { \prime }, cf. Salem number. It seems reasonable to conjecture that S \cup T is closed and that S = T ^ { \prime }, but it is not yet known whether or not T is dense in [ 1 , \infty ).
References
[a1] | M.J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, J.P. Schreiber, "Pisot and Salem Numbers" , Birkhäuser (1992) Zbl 0772.11041 |
[a2] | D.W. Boyd, R.D. Mauldin, "The order type of the set of Pisot numbers" Topology Appl. , 69 (1996) pp. 115–120 |
[a3] | Y. Meyer, "Algebraic numbers and harmonic analysis" , North-Holland (1972) |
[a4] | "The mathematics of long-range aperiodic order" R.V. Moody (ed.) , Kluwer Acad. Publ. (1997) |
[a5] | R. Salem, "Algebraic numbers and Fourier analysis" , Heath (1963) |
[a6] | K. Schmidt, "On periodic expansions of Pisot numbers and Salem numbers" Bull. London Math. Soc. , 12 (1980) pp. 269–278 |
Pisot number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pisot_number&oldid=55297