Pisot–Vijayaraghavan number
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
greater than 1, all of whose
Galois conjugate In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element , over a field extension , are the roots of the minimal polynomial of over . Conjugate elements are commonly called conjug ...
s are less than 1 in
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of
diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by r ...
. They became widely known after the publication of
Charles Pisot Charles Pisot (2 March 1910 – 7 March 1984) was a French mathematician. He is chiefly recognized as one of the primary investigators of the numerical set associated with his name, the Pisot–Vijayaraghavan numbers. He followed the classical p ...
's dissertation in 1938. They also occur in the uniqueness problem for
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
.
Tirukkannapuram Vijayaraghavan Tirukkannapuram Vijayaraghavan ( ta, திருக்கண்ணபுரம் விஜயராகவன்; 30 November 1902 – 20 April 1955) was an Indian mathematician from the Madras region. He worked with G. H. Hardy when he went to ...
and
Raphael Salem Raffaello Sanzio da Urbino, better known as Raphael (; or ; March 28 or April 6, 1483April 6, 1520), was an Italian painter and architect of the High Renaissance. His work is admired for its clarity of form, ease of composition, and visual ...
continued their study in the 1940s.
Salem number In mathematics, a Salem number is a real algebraic integer ''α'' > 1 whose conjugate roots all have absolute value no greater than 1, and at least one of which has absolute value exactly 1. Salem numbers are of interest in Dio ...
s 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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
: \, \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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
. Its minimal element is a cubic irrationality known as the
plastic number In mathematics, the plastic number (also known as the plastic constant, the plastic ratio, the minimal Pisot number, the platin number, Siegel's number or, in French, ) is a mathematical constant which is the unique real solution of the cubic ...
. Much is known about the
accumulation point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s of ''S''. The smallest of them is the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
.


Definition and properties

An algebraic integer of degree ''n'' is a root ''α'' of an irreducible
monic polynomial In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form: :x^n+c_x^+\ ...
''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 Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
numbers of absolute value less than 1, so that they lie strictly inside the circle , ''x'',  = 1 in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, then ''α'' is called a Pisot number, Pisot–Vijayaraghavan number, or simply PV number. For example, the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
, ''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

* 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'', . Consequently, all PV numbers that are less than 2 are algebraic units. * If α is a PV number then so are its powers α''k'', 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 In mathematics, a Perron number is an algebraic integer α which is real and exceeds 1, but such that its conjugate elements are all less than α in absolute value. For example, the larger of the two roots of the irreducible polynomial x^ -3x + ...
(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 \mathbb(\alpha) then the sequence : \, \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 :: \sum_^\infty \, \lambda\alpha^n\, ^2 < \infty. :Then ''α'' is a Pisot number and ''λ'' is an algebraic number in the field \mathbb(\alpha) (Pisot's theorem). * Suppose ''α'' is an algebraic number greater than 1 and ''λ'' is a non-zero real number such that :: \, \lambda\alpha^n\, \to 0, \quad n\to\infty. :Then ''α'' is a Pisot number and ''λ'' is an algebraic number in the field \mathbb(\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 , , ''λα''''n'', , 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 Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
: it contains all its
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s. His proof uses a constructive version of the main diophantine property of Pisot numbers:Salem (1963) p.13 given a Pisot number ''α'', a real number ''λ'' can be chosen so that 0 < ''λ'' ≤ ''α'' and : \sum_^\infty \, \lambda\alpha^n\, ^2 \leq 9. Thus the ''ℓ''2 norm of the sequence , , ''λα''''n'', , 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 In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defin ...
.
Carl Ludwig Siegel Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, ...
proved that it is the positive root of the equation ( 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 derived sets : S, S', S'', \ldots does not terminate. On the other hand, the intersection S^ of these sets is empty, meaning that the Cantor–Bendixson rank of ''S'' is ''ω''. Even more accurately, the
order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y suc ...
of ''S'' has been determined. The set of
Salem number In mathematics, a Salem number is a real algebraic integer ''α'' > 1 whose conjugate roots all have absolute value no greater than 1, and at least one of which has absolute value exactly 1. Salem numbers are of interest in Dio ...
s, 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''.Salem (1963) p.30 It has been conjectured that the union of ''S'' and ''T'' is closed.Salem (1963) p. 31


Quadratic irrationals

If \alpha\, is a
quadratic irrational In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducibl ...
there is only one other conjugate: \alpha'\,, obtained by changing the sign of the square root in \alpha\, from :\alpha = a + \sqrt D\text\alpha' = a - \sqrt D\, or from : \alpha = \frac\text\alpha' = \frac.\, 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 (a-1)^2 or a^2. These are satisfied in the second case exactly when a>0 and either (a-2)^2 or a^2. Thus, the first few quadratic irrationals that are PV numbers are:


Powers of PV-numbers

Pisot–Vijayaraghavan numbers can be used to generate
almost integer In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one. Almost integers are considered interesting when they arise in some context in which they are unexpected. Almost in ...
s: the ''n''th power of a Pisot number approaches integers as ''n'' grows. For example, : (3+\sqrt)^6=27379+8658\sqrt=54757.9999817\dots \approx 54758-\frac. Since 27379 \, and 8658\sqrt \, differ by only 0.0000182\dots,\, : \frac=3.162277662\dots is extremely close to : \sqrt=3.162277660\dots . Indeed : \left( \frac\right)^2=10+\frac. Higher powers give correspondingly better rational approximations. This property stems from the fact that for each ''n'', the sum of ''n''th powers of an algebraic integer ''x'' and its conjugates is exactly an integer; this follows from an application of
Newton's identities In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomia ...
. When ''x'' is a Pisot number, the ''n''th 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 In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
''φ'' 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 : x^6-2x^5+x^4-x^2+x-1, are factors of either : x^n(x^2-x-1) + 1\, or : x^n(x^2-x-1) + (x^2-1).\ The first polynomial is divisible by ''x''2 − 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 ''x''''n'' gives expressions that approach ''x''2 − ''x'' − 1 as ''n'' grows very large and have zeros that
converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) * Converge ICT, internet service provider in the Philippines *CONVERGE CFD s ...
to ''φ''. A complementary pair of polynomials, : x^n(x^2-x-1) - 1 and : x^n(x^2-x-1) - (x^2-1)\, yields Pisot numbers that approach φ from above. Two-dimensional
turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
modeling using
logarithmic spiral A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). Mor ...
chains with
self-similarity __NOTOC__ In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically se ...
defined by a constant scaling factor can be reproduced with some small Pisot numbers.


References

* * Chap. 3. * * * * * * *


External links


''Pisot number''
Encyclopedia of Mathematics * {{DEFAULTSORT:Pisot-Vijayaraghavan Number Algebraic numbers