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Pisot–Vijayaraghavan number
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===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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