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Pisot–Vijayaraghavan number
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==Definition and properties== 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'' > 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.}} ===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 $αζ^{n}$ 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> 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. 31</ref>
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