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''Pisot–Vijayaraghavan number''
 
''Pisot–Vijayaraghavan number''
  
A real algebraic integer (cf. [[Algebraic number|Algebraic number]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p1201301.png" />, all of whose other Galois conjugates have absolute value strictly less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p1201302.png" /> (cf. also [[Galois theory|Galois theory]]). That is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p1201303.png" /> satisfies a polynomial equation of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p1201304.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p1201305.png" /> are integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p1201306.png" /> and the roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p1201307.png" /> other than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p1201308.png" /> all lie in the open unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p1201309.png" />. The set of these numbers is traditionally denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013010.png" />. Every positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013011.png" /> is a Pisot number, but a more interesting example is the [[Golden ratio|golden ratio]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013012.png" />. Every real number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013013.png" /> contains infinitely many Pisot numbers of degree equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013014.png" />, and, in fact, every real number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013015.png" /> can be generated by Pisot numbers, even by Pisot units (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013016.png" />).
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A real [[algebraic integer]] $\theta &gt; 1$, 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 $| x | &lt; 1$. The set of these numbers is traditionally denoted by $S$. Every positive integer $n &gt; 1$ 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013018.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013019.png" />, where here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013020.png" /> denotes the distance from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013021.png" /> to the nearest integer. It is an open question whether this property characterizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013022.png" /> among the real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013023.png" /> (Pisot's conjecture). An important result of Ch. Pisot in this direction is that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013025.png" /> are real numbers for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013028.png" /> [[#References|[a1]]].
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The Pisot numbers have the interesting property that if $0 &lt; \lambda \in \mathbf{Z} ( \theta )$, 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 $\theta &gt; 1$ (Pisot's conjecture). An important result of Ch. Pisot in this direction is that if $\theta &gt; 1$ and $\lambda &gt; 0$ are real numbers for which $\sum _ { n = 0 } ^ { \infty } \| \lambda \theta ^ { n } \| ^ { 2 } &lt; \infty$, 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|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013029.png" />, then the set of powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013030.png" /> is harmonious if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013031.png" /> 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]]].
+
The unusual behaviour of the powers of Pisot numbers leads to applications in [[Harmonic analysis|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 $\theta &gt; 1$, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013032.png" /> is a closed and hence nowhere-dense subset of the real line [[#References|[a5]]]. The derived sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013033.png" /> are all non-empty and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013034.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013035.png" />. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013036.png" /> denotes the set of limit points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013038.png" /> the set of limit points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013039.png" />, etc., cf. also [[Limit point of a set|Limit point of a set]]). The order type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013040.png" /> is described in [[#References|[a2]]]. The smallest elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013043.png" /> are explicitly known [[#References|[a1]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013044.png" /> of Pisot numbers and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013045.png" /> of Salem numbers. It is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013046.png" />, cf. [[Salem number|Salem number]]. It seems reasonable to conjecture that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013047.png" /> is closed and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013048.png" />, but it is not yet known whether or not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013049.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120130/p12013050.png" />.
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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====
 
====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)</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>
+
<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)</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>

Revision as of 08:01, 4 March 2022

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
How to Cite This Entry:
Pisot number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pisot_number&oldid=15094
This article was adapted from an original article by David Boyd (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article