Editing Pisot–Vijayaraghavan number (section)

===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]].