Pisot–Vijayaraghavan number


In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1 all of whose Galois conjugates 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 numbers are a closely related set of numbers.
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 ℓ2, then α is a Pisot number. Building on this characterization of PV numbers, Salem showed that the set S of all PV numbers is closed. Its minimal element is a cubic irrationality known as the plastic number. Much is known about the accumulation points of S. The smallest of them is the golden ratio.

Definition and properties

An algebraic integer of degree n is a root α of an irreducible monic polynomial P of degree n with integer coefficients, its minimal polynomial. The other roots of P are called the conjugates of α. If α > 1 but all other roots of P are real or complex numbers of absolute value less than 1, so that they lie strictly inside the circle |x| = 1 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−1 ≈ −0.618, is less than 1. Therefore, φ is a Pisot number. Its minimal polynomial

Elementary properties

The main interest in PV numbers is due to the fact that their powers have a very "biased" distribution. 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. 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.
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 ||λαn|| to 0 for some auxiliary real λ. It is known that there are only countably many numbers α with this property. The problem is to decide whether any of them is 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: it contains all its limit points. His proof uses a constructive version of the main diophantine property of Pisot numbers: given a Pisot number α, a real number λ can be chosen so that 0 < λ ≤ α and
Thus the ℓ2 norm of the sequence ||λαn|| 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 proved that it is the positive root of the equation and is isolated in S. 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. On the other hand, the intersection of these sets is empty, meaning that the Cantor–Bendixson rank of S is ω. Even more accurately, the order type of S has been determined.
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. It has been conjectured that the union of S and T is closed.

Quadratic irrationals

If is a quadratic irrational there is only one other conjugate:, obtained by changing the sign of the square root in from
or from
Here a and D are integers and in the second case a is odd and D is congruent to 1 modulo 4.
The required conditions are α > 1 and −1 < α' < 1.
These are satisfied in the first case exactly when a > 0 and either or. These are satisfied in the second case exactly when and either or.
Thus, the first few quadratic irrationals that are PV numbers are:
ValueRoot of...Numerical value
1.618033...
2.414213...
2.618033...
2.732050...
3.302775...
3.414213...
3.561552...
3.732050...
3.791287...
4.236067...

Powers of PV-numbers

Pisot–Vijayaraghavan numbers can be used to generate almost integers: the nth power of a Pisot number approaches integers as n approaches infinity. For example,
Since and differ by only
is extremely close to
Indeed
Higher powers give correspondingly better rational approximations.
This property stems from the fact that for each n, the sum of nth powers of an algebraic integer x and its conjugates is exactly an integer; this follows from an application of Newton's identities. When x is a Pisot number, the nth powers of the other conjugates tend to 0 as n tends to infinity. Since the sum is an integer, the distance from xn to the nearest integer tends to 0 at an exponential rate.

Small Pisot numbers

All Pisot numbers that do not exceed the golden ratio φ have been determined by Dufresnoy and Pisot. The table below lists ten smallest Pisot numbers in the increasing order.
ValueRoot of...Root of...
11.3247179572447460260
21.3802775690976141157
31.4432687912703731076
41.4655712318767680267
51.5015948035390873664
61.5341577449142669154
71.5452156497327552432
81.5617520677202972947
91.5701473121960543629
101.5736789683935169887

Since these PV numbers are less than 2, they are all units: their minimal polynomials end in 1 or −1.
The polynomials in this table, with the exception of
are factors of either
or
The first polynomial is divisible by x2 − 1 when n is odd and by x − 1 when n is even. It has one other real zero, which is a PV number. 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.