Editing Pisot–Vijayaraghavan number (section)

{{Short description|Type of algebraic integer}}
{{Use American English|date = March 2019}}
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]].