View source for Pisot–Vijayaraghavan number

{{Short description|Type of algebraic integer}}
In [[mathematics]], a '''Pisot–Vijayaraghavan number''', also called simply a '''Pisot number''' or a '''PV number''', is a [[real number|real]] [[algebraic integer]] greater than 1, all of whose [[Galois conjugate]]s are less than 1 in [[absolute value]]. These numbers were discovered by [[Axel Thue]] in 1912 and rediscovered by [[G. H. Hardy]] in 1919 within the context of [[Diophantine approximation]]. They became widely known after the publication of [[Charles Pisot]]'s dissertation in 1938. They also occur in the uniqueness problem for [[Fourier series]]. [[Tirukkannapuram Vijayaraghavan]] and [[Raphael Salem]] continued their study in the 1940s. [[Salem number]]s are a closely related set of numbers.

A characteristic property of PV numbers is that their powers [[almost integer|approach integers]] at an exponential rate. Pisot [[mathematical proof|proved]] a remarkable [[converse (logic)|converse]]: if ''α''&nbsp;>&nbsp;1 is a real number such that the [[sequence]]

: <math>\|\alpha^n\|</math>

measuring the distance from its consecutive powers to the nearest [[integer]] is [[sequence space|square-summable]], or ''ℓ''<sup>&thinsp;2</sup>, then ''α'' is a Pisot number (and, in particular, algebraic). Building on this characterization of PV numbers, Salem showed that the set ''S'' of all PV numbers is [[closed set|closed]]. Its minimal element is a [[cubic polynomial|cubic]] irrationality known as the [[plastic ratio]]. Much is known about the [[accumulation point]]s of ''S''. The smallest of them is the [[golden ratio]].

==Definition and properties==
[[File:Pisot2 Newton map.svg |thumb|upright=1.33 |[[Newton's method]] for {{math|1=p(z) = z{{isup|4}} − z{{isup|3}} − 1.}} The second Pisot number {{nowrap|{{math|χ}} (right)}} and its conjugates at the nuclei of their [[Attractor#Basins_of_attraction|basins of attraction]] in the complex plane. [[Julia set]] of the Newton map in orange, with unit circle and real curve for reference.]]

An '''algebraic integer''' of degree ''n'' is a root ''α'' of an [[irreducible polynomial|irreducible]] [[monic polynomial]] ''P''(''x'') of [[degree of a polynomial|degree]] ''n'' with integer coefficients, its '''[[minimal polynomial (field theory)|minimal polynomial]]'''. The other roots of ''P''(''x'') are called the '''[[conjugate element (field theory)|conjugates]]''' of ''α''. If ''α''&nbsp;>&nbsp;1 but all other roots of ''P''(''x'') are real or [[complex number|complex]] numbers of absolute value less than 1, so that they lie strictly inside the [[unit circle]] in the [[complex plane]], then ''α'' is called a '''Pisot number''', '''Pisot–Vijayaraghavan number''', or simply '''PV number'''. For example, the [[golden ratio]], ''φ'' ≈ 1.618, is a real [[quadratic integer]] that is greater than 1, while the absolute value of its conjugate, −''φ''<sup>−1</sup> ≈ −0.618, is less than 1. Therefore, ''φ'' is a Pisot number. Its minimal polynomial {{nowrap|is ''x''<sup>2</sup> − ''x'' − 1.}}

{{br}}

===Elementary properties===

* Every integer greater than 1 is a PV number. Conversely, every [[rational number|rational]] PV number is an integer greater than 1.
* If α is an [[irrational number|irrational]] PV number whose minimal polynomial ends in ''k'' then α is greater than |''k''|.
* If α is a PV number then so are its powers α<sup>''k''</sup>, for all positive integer exponents ''k''.
* Every real [[algebraic number field]] '''K''' of degree ''n'' contains a PV number of degree ''n''. This number is a field generator. The set of all PV numbers of degree ''n'' in '''K''' is closed under multiplication.
* Given an upper bound ''M'' and degree ''n'', there are only [[finite set|finitely many]] of PV numbers of degree ''n'' that are less than ''M''.
* Every PV number is a [[Perron number]] (a real algebraic number greater than one all of whose conjugates have smaller absolute value).

===Diophantine properties===

The main interest in PV numbers is due to the fact that their powers have a very "biased" distribution (mod 1). If ''α'' is a PV number and ''λ'' is any algebraic integer in the [[field (mathematics)|field]] <math>\mathbb{Q}(\alpha)</math> then the sequence
: <math>\|\lambda\alpha^n\|,</math>
where ||''x''|| denotes the distance from the real number ''x'' to the nearest integer, approaches 0 at an exponential rate. In particular, it is a square-summable sequence
and its terms converge to 0.

Two converse statements are known: they characterize PV numbers among all real numbers and among the algebraic numbers (but under a weaker Diophantine assumption).

* Suppose ''α'' is a real number greater than 1 and ''λ'' is a non-zero real number such that

:: <math> \sum_{n=1}^\infty \|\lambda\alpha^n\|^2 < \infty. </math>

:Then ''α'' is a Pisot number and ''λ'' is an algebraic number in the field <math>\mathbb{Q}(\alpha)</math> ('''Pisot's theorem''').

* Suppose ''α'' is an algebraic number greater than 1 and ''λ'' is a non-zero real number such that

:: <math> \|\lambda\alpha^n\| \to 0, \quad n\to\infty. </math>

:Then ''α'' is a Pisot number and ''λ'' is an algebraic number in the field <math>\mathbb{Q}(\alpha)</math>.

A longstanding '''Pisot–Vijayaraghavan problem''' asks whether the assumption that ''α'' is algebraic can be dropped from the last statement. If the answer is affirmative, Pisot's numbers would be characterized ''among all real numbers'' by the simple convergence of ||''λα''<sup>''n''</sup>|| to 0 for some auxiliary real ''λ''. It is known that there are only [[countable set|countably many]] numbers ''α'' with this property.<ref>{{cite arXiv |eprint=1401.7588 |last1=Schleischitz |first1=Johannes |title=On the rate of accumulation of $αζ^&#123;n&#125;$ mod 1 to 0 |date=2014 |class=math.NT }}</ref><ref>{{Cite journal |last1=Bertin |first1=M. J. |last2=Decomps-Guilloux |first2=A. |last3=Grandet-Hugot |first3=M. |last4=Pathiaux-Delefosse |first4=M. |last5=Schreiber |first5=J. P. |date=1992 |title=Pisot and Salem Numbers |url=https://link.springer.com/book/10.1007/978-3-0348-8632-1 |journal=SpringerLink |language=en |pages=95 |doi=10.1007/978-3-0348-8632-1|isbn=978-3-0348-9706-8 }}</ref> The problem is to decide whether any of them is [[transcendental number|transcendental]].

===Topological properties===

The set of all Pisot numbers is denoted ''S''. Since Pisot numbers are algebraic, the set ''S'' is countable. Raphael Salem proved that this set is [[closed set|closed]]: it contains all its [[limit point]]s.<ref>{{cite journal | zbl=0063.06657 | last=Salem | first=R. | author-link=Raphaël Salem | title=A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan | journal=Duke Math. J. | volume=11 | pages=103–108 | year=1944 | doi=10.1215/s0012-7094-44-01111-7}}</ref> His proof uses a constructive version of the main diophantine property of Pisot numbers:<ref name=Sal13>Salem (1963) p.13</ref> given a Pisot number ''α'', a real number ''λ'' can be chosen so that 0 < ''λ'' ≤ ''α'' and

: <math>\sum_{n=1}^\infty \|\lambda\alpha^n\|^2 \leq 9.</math>

Thus the ''ℓ''<sup>&thinsp;2</sup> norm of the sequence ||''λα''<sup>''n''</sup>|| can be bounded by a uniform constant independent of ''α''. In the last step of the proof, Pisot's characterization is invoked to conclude that the limit of a sequence of Pisot numbers is itself a Pisot number.

Closedness of ''S'' implies that it has a [[minimal element]]. [[Carl Ludwig Siegel|Carl Siegel]] proved that it is the positive root of the equation {{nowrap|1=''x''<sup>3</sup> − ''x'' − 1 = 0}} ([[plastic ratio|plastic constant]]) and is isolated in ''S''.<ref>{{cite journal | zbl=0063.07005 | last=Siegel | first=Carl Ludwig | author-link=Carl Ludwig Siegel | title=Algebraic integers whose conjugates lie in the unit circle | journal=Duke Math. J. | volume=11 | pages=597–602 | year=1944 | issue=3 | doi=10.1215/S0012-7094-44-01152-X}}</ref> He constructed two sequences of Pisot numbers converging to the golden ratio ''φ'' from below and asked whether ''φ'' is the smallest limit point of ''S''. This was later proved by Dufresnoy and Pisot, who also determined all elements of ''S'' that are less than ''φ''; not all of them belong to Siegel's two sequences. Vijayaraghavan proved that ''S'' has infinitely many limit points; in fact, the sequence of [[derived set (mathematics)|derived set]]s

: <math>S, S', S'', \ldots</math>

does not terminate. On the other hand, the intersection <math>S^{(\omega)}</math> of these sets is [[empty set|empty]], meaning that the [[Derived set (mathematics)#Cantor–Bendixson rank|Cantor–Bendixson rank]] of ''S'' is ''ω''. Even more accurately, the [[order type]] of ''S'' has been determined.<ref>{{Cite journal |last1=Boyd |first1=David W. |author-link=David William Boyd |last2=Mauldin |first2=R. Daniel |title=The Order Type of the Set of Pisot Numbers |journal=Topology and Its Applications |volume=69 |year=1996 |issue=2 |pages=115–120 |doi=10.1016/0166-8641(95)00029-1|doi-access=free }}</ref>

The set of [[Salem number]]s, denoted by ''T'', is intimately related with ''S''.  It has been proved that ''S'' is contained in the set ''T''' of the limit points of ''T''.<ref>{{cite journal | zbl=0060.21601 | last=Salem | first=R. | author-link=Raphaël Salem | title=Power series with integral coefficients | journal=Duke Math. J. | volume=12 | pages=153–172 | year=1945 | doi=10.1215/s0012-7094-45-01213-0}}</ref><ref name=Sal30>Salem (1963) p.30</ref>  It has been [[conjecture]]d that the [[union (set theory)|union]] of ''S'' and ''T'' is closed.<ref name=Sal31>Salem (1963) p.&nbsp;31</ref>

==Quadratic irrationals==

If <math>\alpha\,</math> is a [[quadratic irrational]] there is only one other conjugate, <math>\alpha'</math>, obtained by changing the sign of the [[square root]] in <math>\alpha</math> from

:<math>\alpha = a + \sqrt D \text{ to } \alpha' = a - \sqrt D\, </math>

or from

: <math>\alpha = \frac{a +  \sqrt D}{2}\text{ to }\alpha' = \frac{a -  \sqrt D}{2}.\,</math>

Here ''a'' and ''D'' are integers and in the second case ''a'' is [[parity (mathematics)|odd]] and ''D'' is [[modular arithmetic|congruent]] to 1 modulo 4.

The required conditions are ''α''&nbsp;>&nbsp;1 and −1&nbsp;<&nbsp;''α'''&nbsp;<&nbsp;1.
These are satisfied in the first case exactly when ''a''&nbsp;>&nbsp;0 and either <math>(a-1)^2 < D < a^2</math> or <math>a^2 < D < (a+1)^2</math>, and are satisfied in the second case exactly when <math>a>0</math> and either <math>(a-2)^2 < D < a^2</math> or <math>a^2 < D < (a+2)^2</math>.

Thus, the first few quadratic irrationals that are PV numbers are:

<div class="overflowbugx" style="overflow-x:auto;">
{| class="wikitable"
|-
! Value !! Root of... !! Numerical value
|- style="height:50px"
| <math>\frac{1+\sqrt{5}}{2}</math>  ||<math>x^2-x-1</math>  || 1.618033...  {{OEIS2C|A001622}} (the [[golden ratio]])
|- style="height:50px"
| <math>1+\sqrt{2}\,</math>          ||<math>x^2-2x-1</math> || 2.414213... {{OEIS2C|A014176}} (the [[silver ratio]])
|- style="height:50px"
| <math>\frac{3+\sqrt{5}}{2}</math>  ||<math>x^2-3x+1</math> || 2.618033... {{OEIS2C|A104457}} (the golden ratio squared)
|- style="height:50px"
| <math>1+\sqrt{3}\,</math>          ||<math>x^2-2x-2</math> || 2.732050... {{OEIS2C|A090388}}
|- style="height:50px"
| <math>\frac{3+\sqrt{13}}{2}</math> ||<math>x^2-3x-1</math> || 3.302775... {{OEIS2C|A098316}} (the third [[metallic mean]])
|- style="height:50px"
| <math>2+\sqrt{2}\,</math>          ||<math>x^2-4x+2</math> || 3.414213...
|- style="height:50px"
| <math>\frac{3+\sqrt{17}}{2}</math> ||<math>x^2-3x-2</math> || 3.561552.. {{OEIS2C|A178255}}.
|- style="height:50px"
| <math>2+\sqrt{3}\,</math>          ||<math>x^2-4x+1</math> || 3.732050... {{OEIS2C|A019973}}
|- style="height:50px"
| <math>\frac{3+\sqrt{21}}{2}</math> ||<math>x^2-3x-3</math> || 3.791287...{{OEIS2C|A090458}}
|- style="height:50px"
| <math>2+\sqrt{5}\,</math>          ||<math>x^2-4x-1</math> || 4.236067... {{OEIS2C|A098317}} (the fourth metallic mean)
|}
</div>

==Powers of PV-numbers==

Pisot–Vijayaraghavan numbers can be used to generate [[almost integer]]s: the ''n''th power of a Pisot number approaches integers as ''n'' grows. For example,

: <div class="overflowbugx" style="overflow-x:auto;"><math>(3+\sqrt{10})^6=27379+8658\sqrt{10}=54757.9999817\dots \approx 54758-\frac{1}{54758}.</math></div>

Since <math> 27379 \, </math> and <math> 8658\sqrt{10} \, </math> differ by only <math> 0.0000182\dots,\, </math>

: <math>\frac{27379}{8658}=3.162277662\dots</math>

is extremely close to

: <math>\sqrt{10}=3.162277660\dots .</math>

Indeed

: <math>\left( \frac{27379}{8658}\right)^2=10+\frac{1}{8658^2}.</math>

Higher powers give correspondingly better rational approximations.

This property stems from the fact that for each ''n'', the sum of ''n''th powers of an algebraic integer ''x'' and its conjugates is exactly an integer; this follows from an application of [[Newton's identities]]. When ''x'' is a Pisot number, the ''n''th powers of the other conjugates tend to 0 as ''n'' tends to infinity. Since the sum is an integer, the distance from ''x<sup>n</sup>'' to the nearest integer tends to 0 at an exponential rate.

==Small Pisot numbers==

All Pisot numbers that do not exceed the [[golden ratio]] ''φ'' have been determined by Dufresnoy and Pisot. The table below lists ten smallest Pisot numbers in increasing order.<ref>{{citation
 | last1 = Dufresnoy | first1 = J.
 | last2 = Pisot | first2 = Ch.
 | journal = Annales Scientifiques de l'École Normale Supérieure
 | language = French
 | mr = 0072902
 | pages = 69–92
 | title = Etude de certaines fonctions méromorphes bornées sur le cercle unité. Application à un ensemble fermé d'entiers algébriques
 | url = http://www.numdam.org/item?id=ASENS_1955_3_72_1_69_0
 | volume = 72
 | year = 1955| doi = 10.24033/asens.1030
 }}. The smallest of these numbers are listed in numerical order on p. 92.</ref>

<div class="overflowbugx" style="overflow-x:auto;">
{| class="wikitable"
|-
!   !! Value !! Root of... !! Root of...
|-
| 1 || {{val|1.32471795724474602596|end=...}} {{OEIS2C|A060006}} ([[plastic ratio]]) || <math>x(x^2-x-1)+(x^2-1)</math> || <math>x^3-x-1</math>
|-
| 2 || {{val|1.38027756909761411567|end=...}} {{OEIS2C|A086106}} || <math>x^2(x^2-x-1)+(x^2-1)</math> ||<math>x^4-x^3-1</math>
|-
| 3 || {{val|1.44326879127037310762|end=...}} {{OEIS2C|A228777}} || <math>x^3(x^2-x-1)+(x^2-1)</math> ||<math>x^5-x^4-x^3+x^2-1</math>
|-
| 4 || {{val|1.46557123187676802665|end=...}} {{OEIS2C|A092526}} ([[supergolden ratio]]) || <math>x^3(x^2-x-1)+1</math> ||<math>x^3-x^2-1</math>
|-
| 5 || {{val|1.50159480353908736637|end=...}} {{OEIS2C|A293508}} || <math>x^4(x^2-x-1)+(x^2-1)</math> ||<math>x^6-x^5-x^4+x^2-1</math>
|-
| 6 || {{val|1.53415774491426691543|end=...}} {{OEIS2C|A293509}} || <math>x^4(x^2-x-1)+1</math> ||<math>x^5-x^3-x^2-x-1</math>
|-
| 7 || {{val|1.54521564973275524325|end=...}} {{OEIS2C|A293557}} || <math>x^5(x^2-x-1)+(x^2-1)</math> ||<math>x^7-x^6-x^5+x^2-1</math>
|-
| 8 || {{val|1.56175206772029729470|end=...}} {{OEIS2C|A374002}} || <math>x^3(x^3-2x^2+x-1)+(x-1)(x^2+1)</math> ||<math>x^6-2x^5+x^4-x^2+x-1</math>
|-
| 9 || {{val|1.57014731219605436291|end=...}} {{OEIS2C|A293506}} || <math>x^5(x^2-x-1)+1</math> ||<math>x^5-x^4-x^2-1</math>
|-
| 10 || {{val|1.57367896839351698877|end=...}} {{OEIS2C|A374003}} || <math>x^6(x^2-x-1)+(x^2-1)</math> ||<math>x^8-x^7-x^6+x^2-1</math>
|}
</div>

Since these PV numbers are less than 2, they are all units: their minimal polynomials end in 1 or −1.
<!-- this needs to be put in proper place or removed
Every real algebraic number field contains a PV number that generates this field. In real quadratic and cubic fields it is not hard to find a unit that is a PV number.
-->
The polynomials in this table,<ref name=":0">Bertin et al., p. 133.</ref> with the exception of

: <math>x^6 - 2x^5 + x^4 - x^2 + x - 1,</math>

are factors of either

:<math>x^n(x^2 - x - 1) + 1</math>
or
:<math>x^n(x^2 - x - 1) + (x^2 - 1).</math>

The first polynomial is divisible by ''x''<sup>2</sup>&nbsp;−&nbsp;1 when ''n'' is odd and by ''x''&nbsp;−&nbsp;1 when ''n'' is [[parity (mathematics)|even]]. It has one other real zero, which is a PV number. Dividing either polynomial by ''x''<sup>''n''</sup> gives expressions that approach ''x''<sup>2</sup>&nbsp;−&nbsp;''x''&nbsp;−&nbsp;1 as ''n'' grows very large and have zeros that [[limit of a sequence|converge]] to ''φ''.  A complementary pair of polynomials,

:<math>x^n(x^2 - x - 1) - 1</math>
and
:<math> x^n(x^2-x-1) - (x^2-1)\,</math>

yields Pisot numbers that approach φ from above.

Two-dimensional [[turbulence]] modeling using [[logarithmic spiral]] chains with [[self-similarity]] defined by a constant scaling factor can be reproduced with some small Pisot numbers.<ref>{{cite journal |url=https://journals.aps.org/pre/abstract/10.1103/PhysRevE.100.043113 |author1=Ö. D. Gürcan |author2= Shaokang Xu|author3=P. Morel |title=Spiral chain models of two-dimensional turbulence |journal=Physical Review E |volume=100 |date=2019 |issue=4 |page=043113 |doi=10.1103/PhysRevE.100.043113 |pmid=31770954 |arxiv=1903.09494 }}</ref>

==References==
{{Reflist}}
* {{cite book | author=M.J. Bertin |author2=A. Decomps-Guilloux |author3=M. Grandet-Hugot |author4=M. Pathiaux-Delefosse |author5=J.P. Schreiber | title=Pisot and Salem Numbers | publisher=Birkhäuser | year=1992 | isbn=3-7643-2648-4}}
* {{cite book | first=Peter|last= Borwein | author-link=Peter Borwein | title=Computational Excursions in Analysis and Number Theory | series=CMS Books in Mathematics | publisher=[[Springer-Verlag]] | year=2002 | isbn=0-387-95444-9 | zbl=1020.12001}} Chap. 3.
* {{cite journal | last=Boyd | first=David W. | title=Pisot and Salem numbers in intervals of the real line | journal=Math. Comp. | volume=32 | year=1978 | issue=144 | pages=1244–1260 | doi=10.2307/2006349 | zbl=0395.12004 | issn=0025-5718 | doi-access=free| jstor=2006349 }}
* {{cite book | first1=J. W. S. |last1=Cassels | author-link=J. W. S. Cassels | title=An introduction to Diophantine approximation | series=Cambridge Tracts in Mathematics and Mathematical Physics | volume=45 | publisher=[[Cambridge University Press]] | year=1957 | pages=133–144}}
* {{cite journal | first=G. H. |last= Hardy | author-link=G. H. Hardy | title=A problem of diophantine approximation | journal=J. Indian Math. Soc. | volume=11 | year=1919 | pages=205–243}}
* {{cite journal |author1=Ö. D. Gürcan |author2= Shaokang Xu |author3=P. Morel |title=Spiral chain models of two-dimensional turbulence |journal=Physical Review E |volume=100 |year=2019 |issue= 4 |page= 043113 |doi=10.1103/PhysRevE.100.043113 |pmid= 31770954 |arxiv=1903.09494}}
* {{cite journal | first = Charles|last= Pisot | author-link=Charles Pisot | title=La répartition modulo 1 et nombres algébriques | journal=Ann. Sc. Norm. Super. Pisa II|series= Ser. 7 | year=1938 | pages=205–248 | zbl=0019.15502 | language=French}}
* {{cite book | last=Salem | first=Raphaël | author-link=Raphaël Salem | title=Algebraic numbers and Fourier analysis | series=Heath mathematical monographs | location=Boston, MA | publisher=[[D. C. Heath and Company]] | year=1963 | zbl=0126.07802}}
* {{cite journal | first=Axel | last=Thue | author-link=Axel Thue | title=Über eine Eigenschaft, die keine transzendente Größe haben kann | journal=Christiania Vidensk. Selsk. Skrifter | volume=2 | number=20 | year=1912 | pages=1–15 | jfm=44.0480.04}}

==External links==
* [https://encyclopediaofmath.org/wiki/Pisot_number ''Pisot number''], Encyclopedia of Mathematics
* {{MathWorld|urlname=PisotNumber|title=Pisot Number|author=Terr, David|author2=Weisstein, Eric W.|name-list-style=amp}}

{{Algebraic numbers}}
{{Metallic ratios}}

{{DEFAULTSORT:Pisot-Vijayaraghavan Number}}
[[Category:Algebraic numbers]]