In mathematics, an irrationality measure of a real number is a measure of how "closely" it can be approximated by rationals.
If a function , defined for , takes positive real values and is strictly decreasing in both variables, consider the following inequality:
for a given real number and rational numbers with . Define as the set of all for which only finitely many exist, such that the inequality is satisfied. Then is called an irrationality measure of with regard to If there is no such and the set is empty, is said to have infinite irrationality measure .
Consequently the inequality
has at most only finitely many solutions for all . [1]
The irrationality exponent or Liouville–Roth irrationality measure is given by setting , [1] a definition adapting the one of Liouville numbers — the irrationality exponent is defined for real numbers to be the supremum of the set of such that is satisfied by an infinite number of coprime integer pairs with . [2] [3] : 246
For any value , the infinite set of all rationals satisfying the above inequality yields good approximations of . Conversely, if , then there are at most finitely many coprime with that satisfy the inequality.
For example, whenever a rational approximation with yields exact decimal digits, then
for any , except for at most a finite number of "lucky" pairs .
A number with irrationality exponent is called a diophantine number, [4] while numbers with are called Liouville numbers.
Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2.
On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers, have an irrationality exponent exactly equal to 2. [3] : 246
It is for real numbers and rational numbers and . If for some we have , then it follows . [5] : 368
For a real number given by its simple continued fraction expansion with convergents it holds: [1]
If we have and for some positive real numbers , then we can establish an upper bound for the irrationality exponent of by: [6] [7]
For most transcendental numbers, the exact value of their irrationality exponent is not known. [5] Below is a table of known upper and lower bounds.
Number | Irrationality exponent | Notes | |
---|---|---|---|
Lower bound | Upper bound | ||
Rational number with | 1 | Every rational number has an irrationality exponent of exactly 1. | |
Irrational algebraic number | 2 | By Roth's theorem the irrationality exponent of any irrational algebraic number is exactly 2. Examples include square roots and the golden ratio . | |
2 | If the elements of the simple continued fraction expansion of an irrational number are bounded above by an arbitrary polynomial , then its irrationality exponent is . Examples include numbers which continued fractions behave predictably such as and . | ||
2 | |||
2 | |||
with | 2 | with , has continued fraction terms which do not exceed a fixed constant. [8] [9] | |
with [10] | 2 | where is the Thue–Morse sequence and . See Prouhet-Thue-Morse constant. | |
[11] [12] | 2 | 3.57455... | There are other numbers of the form for which bounds on their irrationality exponents are known. [13] [14] [15] |
[11] [16] | 2 | 5.11620... | |
[17] | 2 | 3.43506... | There are many other numbers of the form for which bounds on their irrationality exponents are known. [17] This is the case for . |
[18] [19] | 2 | 4.60105... | There are many other numbers of the form for which bounds on their irrationality exponents are known. [18] This is the case for . |
[11] [20] | 2 | 7.10320... | It has been proven that if the Flint Hills series (where n is in radians) converges, then 's irrationality exponent is at most [21] [22] and that if it diverges, the irrationality exponent is at least . [23] |
[11] [24] | 2 | 5.09541... | and are linearly dependent over . |
[25] | 2 | 9.27204... | There are many other numbers of the form for which bounds on their irrationality exponents are known. [26] [27] |
[28] | 2 | 5.94202... | |
Apéry's constant [11] | 2 | 5.51389... | |
[29] | 2 | 10330 | |
Cahen's constant [30] | 3 | ||
Champernowne constants in base [31] | Examples include | ||
Liouville numbers | The Liouville numbers are precisely those numbers having infinite irrationality exponent. [3] : 248 |
The irrationality base or Sondow irrationality measure is obtained by setting . [1] [6] It is a weaker irrationality measure, being able to distinguish how well different Liouville numbers can be approximated, but yielding for all other real numbers:
Let be an irrational number. If there exist real numbers with the property that for any , there is a positive integer such that
for all integers with then the least such is called the irrationality base of and is represented as .
If no such exists, then and is called a super Liouville number.
If a real number is given by its simple continued fraction expansion with convergents then it holds:
Any real number with finite irrationality exponent has irrationality base , while any number with irrationality base has irrationality exponent and is a Liouville number.
The number has irrationality exponent and irrationality base .
The numbers ( represents tetration, ) have irrationality base .
The number has irrationality base , hence it is a super Liouville number.
Although it is not known whether or not is a Liouville number, [32] : 20 it is known that . [5] : 371
Setting gives a stronger irrationality measure: the Markov constant . For an irrational number it is the factor by which Dirichlet's approximation theorem can be improved for . Namely if is a positive real number, then the inequality
has infinitely many solutions . If there are at most finitely many solutions.
Dirichlet's approximation theorem implies and Hurwitz's theorem gives both for irrational . [33]
This is in fact the best general lower bound since the golden ratio gives . It is also .
Given by its simple continued fraction expansion, one may obtain: [34]
Bounds for the Markov constant of can also be given by with . [35] This implies that if and only if is not bounded and in particular if is a quadratic irrational number. A further consequence is .
Any number with or has an unbounded simple continued fraction and hence .
The values and imply that the inequality has for all infinitely many solutions while the inequality has for all only at most finitely many solutions . This gives rise to the question what the best upper bound is. The answer is given by: [36]
which is satisfied by infinitely many for but not for .
This makes the number alongside the rationals and quadratic irrationals an exception to the fact that for almost all real numbers the inequality below has infinitely many solutions : [5] (see Khinchin's theorem)
Kurt Mahler extended the concept of an irrationality measure and defined a so-called transcendence measure, drawing on the idea of a Liouville number and partitioning the transcendental numbers into three distinct classes. [3]
Instead of taking for a given real number the difference with , one may instead focus on term with and with . Consider the following inequality:
with and .
Define as the set of all for which infinitely many solutions exist, such that the inequality is satisfied. Then is Mahler's irrationality measure. It gives for rational numbers, for algebraic irrational numbers and in general , where denotes the irrationality exponent.
Mahler's irrationality measure can be generalized as follows: [2] [3] Take to be a polynomial with and integer coefficients . Then define a height function and consider for complex numbers the inequality:
with .
Set to be the set of all for which infinitely many such polynomials exist, that keep the inequality satisfied. Further define for all with being the above irrationality measure, being a non-quadraticity measure, etc.
Then Mahler's transcendence measure is given by:
The transcendental numbers can now be divided into the following three classes:
If for all the value of is finite and is finite as well, is called an S-number (of type ).
If for all the value of is finite but is infinite, is called an T-number.
If there exists a smallest positive integer such that for all the are infinite, is called an U-number (of degree ).
The number is algebraic (and called an A-number) if and only if .
Almost all numbers are S-numbers. In fact, almost all real numbers give while almost all complex numbers give . [37] : 86 The number e is an S-number with . The number π is either an S- or T-number. [37] : 86 The U-numbers are a set of measure 0 but still uncountable. [38] They contain the Liouville numbers which are exactly the U-numbers of degree one.
Another generalization of Mahler's irrationality measure gives a linear independence measure. [2] [13] For real numbers consider the inequality
with and .
Define as the set of all for which infinitely many solutions exist, such that the inequality is satisfied. Then is the linear independence measure.
If the are linearly dependent over then .
If are linearly independent algebraic numbers over then . [32]
It is further .
Jurjen Koksma in 1939 proposed another generalization, similar to that of Mahler, based on approximations of complex numbers by algebraic numbers. [3] [37]
For a given complex number consider algebraic numbers of degree at most . Define a height function , where is the characteristic polynomial of and consider the inequality:
with .
Set to be the set of all for which infinitely many such algebraic numbers exist, that keep the inequality satisfied. Further define for all with being an irrationality measure, being a non-quadraticity measure, [17] etc.
Then Koksma's transcendence measure is given by:
The complex numbers can now once again be partitioned into four classes A*, S*, T* and U*. However it turns out that these classes are equivalent to the ones given by Mahler in the sense that they produce exactly the same partition. [37] : 87
Given a real number , an irrationality measure of quantifies how well it can be approximated by rational numbers with denominator . If is taken to be an algebraic number that is also irrational one may obtain that the inequality
has only at most finitely many solutions for . This is known as Roth's theorem.
This can be generalized: Given a set of real numbers one can quantify how well they can be approximated simultaneously by rational numbers with the same denominator . If the are taken to be algebraic numbers, such that are linearly independent over the rational numbers it follows that the inequalities
have only at most finitely many solutions for . This result is due to Wolfgang M. Schmidt. [39] [40]
In mathematics, the infimum of a subset of a partially ordered set is the greatest element in that is less than or equal to each element of if such an element exists. If the infimum of exists, it is unique, and if b is a lower bound of , then b is less than or equal to the infimum of . Consequently, the term greatest lower bound is also commonly used. The supremum of a subset of a partially ordered set is the least element in that is greater than or equal to each element of if such an element exists. If the supremum of exists, it is unique, and if b is an upper bound of , then the supremum of is less than or equal to b. Consequently, the supremum is also referred to as the least upper bound.
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz.
In number theory, a Liouville number is a real number with the property that, for every positive integer , there exists a pair of integers with such that
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non-increasing, or non-decreasing. In its simplest form, it says that a non-decreasing bounded-above sequence of real numbers converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums if and only if the partial sums are bounded.
In probability theory, the law of large numbers (LLN) is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the LLN states that given a sample of independent and identically distributed values, the sample mean converges to the true mean.
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces.
In information theory, the asymptotic equipartition property (AEP) is a general property of the output samples of a stochastic source. It is fundamental to the concept of typical set used in theories of data compression.
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation.
In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded in absolute value by an integrable function and is almost everywhere point wise convergent to a function then the sequence converges in to its point wise limit, and in particular the integral of the limit is the limit of the integrals. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration.
In number theory, Khinchin's constant is a mathematical constant related to the simple continued fraction expansions of many real numbers. In particular Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, the coefficients ai of the continued fraction expansion of x have a finite geometric mean that is independent of the value of x. It is known as Khinchin's constant and denoted by K0.
In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b]. Functions whose total variation is finite are called functions of bounded variation.
In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin.
In measure theory, Carathéodory's extension theorem states that any pre-measure defined on a given ring of subsets R of a given set Ω can be extended to a measure on the σ-ring generated by R, and this extension is unique if the pre-measure is σ-finite. Consequently, any pre-measure on a ring containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers. This is an extremely powerful result of measure theory, and leads, for example, to the Lebesgue measure.
In mathematics, a π-system on a set is a collection of certain subsets of such that
An -superprocess, , within mathematics probability theory is a stochastic process on that is usually constructed as a special limit of near-critical branching diffusions.
In probability theory, an -divergence is a certain type of function that measures the difference between two probability distributions and . Many common divergences, such as KL-divergence, Hellinger distance, and total variation distance, are special cases of -divergence.
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line which consists of the real numbers and
In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s, are generalisations of the more familiar spaces.
In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The entropic value at risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid, which is an upper bound for the value at risk (VaR) and the conditional value at risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy. Because of its connection with the VaR and the relative entropy, this risk measure is called "entropic value at risk". The EVaR was developed to tackle some computational inefficiencies of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid developed a wide class of coherent risk measures, called g-entropic risk measures. Both the CVaR and the EVaR are members of this class.
A generative adversarial network (GAN) is a class of machine learning frameworks and a prominent framework for approaching generative artificial intelligence. The concept was initially developed by Ian Goodfellow and his colleagues in June 2014. In a GAN, two neural networks contest with each other in the form of a zero-sum game, where one agent's gain is another agent's loss.