Khinchin's constant
Encyclopedia
In number theory
, Aleksandr Yakovlevich Khinchin
proved that for almost all
real numbers x, coefficients ai of the continued fraction
expansion of x have a finite geometric mean
that is independent of the value of x and is known as Khinchin's constant.
That is, for
it is almost always
true that
where
is Khinchin's constant
Among the numbers x whose continued fraction expansions do not have this property are rational number
s, solutions of quadratic equation
s with rational coefficients (including the golden ratio
φ), and the base of the natural logarithm
e.
Among the numbers whose continued fraction expansions apparently do have this property (based on numerical evidence) are π
, the Euler–Mascheroni constant
γ, and Khinchin's constant itself. However, this is unproven, because even though it is known that almost all
real numbers have this property, it has not been proven for any specific real number whose full continued fraction representation is not known.
Khinchin is sometimes spelled Khintchine (the French transliteration) in older mathematical literature.
.
Since the first coefficient a0 of the continued fraction of x plays no role in Khinchin's theorem and since the rational numbers have Lebesgue measure
zero, we are reduced to the study of irrational numbers in the unit interval
, i.e., those in
. These numbers are in bijection
with infinite continued fraction
s of the form [0; a1, a2, ...], which we simply write [a1, a2, ...], where a1, a2, ... are positive integers. Define a transformation T:I → I by
The transformation T is called the Gauss–Kuzmin–Wirsing operator. For every Borel subset
E of I, we also define the Gauss–Kuzmin measure of E
Then μ is a probability measure
on the σ-algebra of Borel subsets of I. The measure μ is equivalent
to the Lebesgue measure on I, but it has the additional property that the transformation T preserves the measure μ. Moreover, it can be proved that T is an ergodic transformation of the measurable space I endowed with the probability measure μ (this is the hard part of the proof). The ergodic theorem then says that for any μ-integrable function f on I, the average value of
is the same for almost all 
:
Applying this to the function defined by f([a1, a2, ...]) = log(a1), we obtain that
for almost all [a1, a2, ...] in I as n → ∞.
Taking the exponential
on both sides, we obtain to the left the geometric mean
of the first n coefficients of the continued fraction, and to the right Khinchin's constant.
in the form
or, by peeling off terms in the series,
where N is an integer, held fixed, and ζ(s, n) is the Hurwitz zeta function. Both series are strongly convergent, as ζ(n) − 1 approaches zero quickly for large n. An expansion may also be given in terms of the dilogarithm:
When the {an} are the terms of a continued fraction expansion, the constants are given by
This is obtained by taking the p-th mean in conjunction with the Gauss–Kuzmin distribution. The value for K0 may be shown to be obtained in the limit of p → 0.
of the terms of a continued fraction may be obtained as well. The value obtained is
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, Aleksandr Yakovlevich Khinchin
Aleksandr Yakovlevich Khinchin
Aleksandr Yakovlevich Khinchin was a Soviet mathematician and one of the most significant people in the Soviet school of probability theory. He was born in the village of Kondrovo, Kaluga Governorate, Russian Empire. While studying at Moscow State University, he became one of the first followers...
proved that for almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
real numbers x, coefficients ai of the continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
expansion of x have a finite geometric mean
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
that is independent of the value of x and is known as Khinchin's constant.
That is, for
it is almost always
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
true that
where
is Khinchin's constant Among the numbers x whose continued fraction expansions do not have this property are rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s, solutions of quadratic equation
Quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
s with rational coefficients (including the golden ratio
Golden ratio
In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
φ), and the base of the natural logarithm
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
e.
Among the numbers whose continued fraction expansions apparently do have this property (based on numerical evidence) are π
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
, the Euler–Mascheroni constant
Euler–Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
γ, and Khinchin's constant itself. However, this is unproven, because even though it is known that almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
real numbers have this property, it has not been proven for any specific real number whose full continued fraction representation is not known.
Khinchin is sometimes spelled Khintchine (the French transliteration) in older mathematical literature.
Sketch of proof
The proof presented here was arranged by and is much simpler than Khinchin's original proof which did not use ergodic theoryErgodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
.
Since the first coefficient a0 of the continued fraction of x plays no role in Khinchin's theorem and since the rational numbers have Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
zero, we are reduced to the study of irrational numbers in the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
, i.e., those in
. These numbers are in bijectionBijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
with infinite continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
s of the form [0; a1, a2, ...], which we simply write [a1, a2, ...], where a1, a2, ... are positive integers. Define a transformation T:I → I by
The transformation T is called the Gauss–Kuzmin–Wirsing operator. For every Borel subset
Borel set
In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
E of I, we also define the Gauss–Kuzmin measure of E
Then μ is a probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
on the σ-algebra of Borel subsets of I. The measure μ is equivalent
Equivalence (measure theory)
In mathematics, and specifically in measure theory, equivalence is a notion of two measures being "the same".-Definition:Let be a measurable space, and let μ, ν : Σ → R be two signed measures. Then μ is said to be equivalent to ν if and only if each is absolutely continuous with respect to the other...
to the Lebesgue measure on I, but it has the additional property that the transformation T preserves the measure μ. Moreover, it can be proved that T is an ergodic transformation of the measurable space I endowed with the probability measure μ (this is the hard part of the proof). The ergodic theorem then says that for any μ-integrable function f on I, the average value of
is the same for almost all :Applying this to the function defined by f([a1, a2, ...]) = log(a1), we obtain that
for almost all [a1, a2, ...] in I as n → ∞.
Taking the exponential
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
on both sides, we obtain to the left the geometric mean
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...
of the first n coefficients of the continued fraction, and to the right Khinchin's constant.
Series expressions
Khinchin's constant may be expressed as a rational zeta seriesRational zeta series
In mathematics, a rational zeta series is the representation of an arbitrary real number in terms of a series consisting of rational numbers and the Riemann zeta function or the Hurwitz zeta function...
in the form
or, by peeling off terms in the series,
where N is an integer, held fixed, and ζ(s, n) is the Hurwitz zeta function. Both series are strongly convergent, as ζ(n) − 1 approaches zero quickly for large n. An expansion may also be given in terms of the dilogarithm:
Hölder means
The Khinchin constant can be viewed as the first in a series of the Hölder means of the terms of continued fractions. Given an arbitrary series {an}, the Hölder mean of order p of the series is given byWhen the {an} are the terms of a continued fraction expansion, the constants are given by
This is obtained by taking the p-th mean in conjunction with the Gauss–Kuzmin distribution. The value for K0 may be shown to be obtained in the limit of p → 0.
Harmonic mean
By means of the above expressions, the harmonic meanHarmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....
of the terms of a continued fraction may be obtained as well. The value obtained is