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
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). 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
Definition and properties
An algebraic integer of degree n is a root α of an
irreduciblemonic polynomialP(x) of degree n with integer coefficients, its minimal polynomial. The other roots of P(x) are called the conjugates of α. If α > 1 but all other roots of P(x) 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 is x2 − x − 1.
Every integer greater than 1 is a PV number. Conversely, every rational PV number is an integer greater than 1.
If α is an irrational PV number whose minimal polynomial ends in k then α is greater than |k|.
If α is a PV number then so are its powers αk, for all natural number exponents k.
Every real algebraic number field K of degree n contains a PV number of degree n. This number is a field generator. The set of all PV numbers of degree n in K is closed under multiplication.
Given an upper bound M and degree n, there are only a finite number of PV numbers of degree n that are less than M.
Every PV number is a
Perron number (a real algebraic number greater than one all of whose conjugates have smaller absolute value).
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. 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
Then α is a Pisot number and λ is an algebraic number in the field (Pisot's theorem).
Suppose α is an algebraic number greater than 1 and λ is a non-zero real number such that
Then α is a Pisot number and λ is an algebraic number in the field .
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.
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
Carl Ludwig Siegel proved that it is the positive root of the equation x3 − x − 1 = 0 (
plastic constant) 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
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.
If is a
quadratic irrational there is only one other conjugate: , obtained by changing the sign of the square root in 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:
Pisot–Vijayaraghavan numbers can be used to generate
almost integers: the nth power of a Pisot number approaches integers as n grows. For example,
Since and differ by only
is extremely close to
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 ratioφ have been determined by Dufresnoy and Pisot. The table below lists ten smallest Pisot numbers in the increasing order.
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
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,