In mathematics, a Pisot–Vijayaraghavan number (or Pisot number or 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
‖ α n ‖ {\displaystyle \|\alpha ^{n}\|}
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 ratio. 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(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 unit circle 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.
Elementary properties 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 positive integer 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 finitely many 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).
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 Q ( α ) {\displaystyle \mathbb {Q} (\alpha )} then the sequence
‖ λ α n ‖ , {\displaystyle \|\lambda \alpha ^{n}\|,}
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
∑ n = 1 ∞ ‖ λ α n ‖ 2 < ∞ . {\displaystyle \sum _{n=1}^{\infty }\|\lambda \alpha ^{n}\|^{2}<\infty .}
Then α is a Pisot number and λ is an algebraic number in the field Q ( α ) {\displaystyle \mathbb {Q} (\alpha )} (Pisot's theorem). Suppose α is an algebraic number greater than 1 and λ is a non-zero real number such that
‖ λ α n ‖ → 0 , n → ∞ . {\displaystyle \|\lambda \alpha ^{n}\|\to 0,\quad n\to \infty .}
Then α is a Pisot number and λ is an algebraic number in the field Q ( α ) {\displaystyle \mathbb {Q} (\alpha )} . 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
∑ n = 1 ∞ ‖ λ α n ‖ 2 ≤ 9. {\displaystyle \sum _{n=1}^{\infty }\|\lambda \alpha ^{n}\|^{2}\leq 9.}
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