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''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|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|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 $\lambda \in \mathbf{Q} ( \theta )$ [[#References|[a1]]].

The unusual behaviour of the powers of Pisot numbers leads to applications in [[harmonic analysis]], [[#References|[a3]]], [[#References|[a5]]], dynamical systems theory (cf. also [[Dynamical system|Dynamical system]]) [[#References|[a6]]] and the theory of quasi-crystals [[#References|[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|Salem number]] [[#References|[a3]]]. The Bragg spectrum of the diffraction pattern of a self-similar tiling (cf., e.g., [[Voronoi lattice types|Voronoi lattice types]]) is non-trivial if and only if the scaling factor of the tiling is a Pisot number [[#References|[a4]]].

A surprising fact is that $S$ is a closed and hence nowhere-dense subset of the real line [[#References|[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|Limit point of a set]]). The order type of $S$ is described in [[#References|[a2]]]. The smallest elements of $S$, $S ^ { \prime }$ and $S ^ { \prime \prime }$ are explicitly known [[#References|[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|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====
<table>
<tr><td valign="top">[a1]</td> <td valign="top">  M.J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, J.P. Schreiber, "Pisot and Salem Numbers" , Birkhäuser  (1992) {{ZBL|0772.11041}}</td></tr>
<tr><td valign="top">[a2]</td> <td valign="top">  D.W. Boyd, R.D. Mauldin,   "The order type of the set of Pisot numbers"  ''Topology Appl.'' , '''69'''  (1996)  pp. 115–120</td></tr>
<tr><td valign="top">[a3]</td> <td valign="top">  Y. Meyer, "Algebraic numbers and harmonic analysis" , North-Holland  (1972)</td></tr>
<tr><td valign="top">[a4]</td> <td valign="top">  "The mathematics of long-range aperiodic order"  R.V. Moody (ed.) , Kluwer Acad. Publ.  (1997)</td></tr>
<tr><td valign="top">[a5]</td> <td valign="top">  R. Salem, "Algebraic numbers and Fourier analysis" , Heath  (1963)</td></tr>
<tr><td valign="top">[a6]</td> <td valign="top">  K. Schmidt, "On periodic expansions of Pisot numbers and Salem numbers"  ''Bull. London Math. Soc.'' , '''12'''  (1980)  pp. 269–278</td></tr>
</table>