Pisot number

Pisot–Vijayaraghavan number

A real algebraic integer $\theta > 1$, 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)
[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