These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of Diophantine approximation.
Tirukkannapuram Vijayaraghavan and Raphael Salem continued their study in the 1940s.
A characteristic property of PV numbers is that their powers approach integers at an exponential rate.
Pisot proved a remarkable converse: if α > 1 is a real number such that the sequence measuring the distance from its consecutive powers to the nearest integer is square-summable, or ℓ 2, then α is a Pisot number (and, in particular, algebraic).
Its minimal element is a cubic irrationality known as the plastic ratio.
An algebraic integer of degree n is a root α of an irreducible monic polynomial P(x) of degree n with integer coefficients, its minimal polynomial.
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−1 ≈ −0.618, is less than 1.
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
then the sequence where ||x|| denotes the distance from the real number x to the nearest integer, approaches 0 at an exponential rate.
A longstanding Pisot–Vijayaraghavan problem asks whether the assumption that α is algebraic can be dropped from the last statement.
Raphael Salem proved that this set is closed: it contains all its limit points.
[3] His proof uses a constructive version of the main diophantine property of Pisot numbers:[4] given a Pisot number α, a real number λ can be chosen so that 0 < λ ≤ α and Thus the ℓ 2 norm of the sequence ||λαn|| can be bounded by a uniform constant independent of α.
Carl Siegel proved that it is the positive root of the equation x3 − x − 1 = 0 (plastic constant) and is isolated in S.[5] 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 sets does not terminate.
[6] The set of Salem numbers, 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.[7][8] It has been conjectured that the union of S and T is closed.
is extremely close to Indeed Higher powers give correspondingly better rational approximations.
The table below lists ten smallest Pisot numbers in increasing order.
[10] Since these PV numbers are less than 2, they are all units: their minimal polynomials end in 1 or −1.
Dividing either polynomial by xn gives expressions that approach x2 − x − 1 as n grows very large and have zeros that converge to φ.
A complementary pair of polynomials, and 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.