Khinchin's constant

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 a_i 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.

That is, for

${\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots }}}}}}}}\;}$

it is almost always true that

${\displaystyle \lim _{n\rightarrow \infty }\left(a_{1}a_{2}...a_{n}\right)^{1/n}=K_{0}.}$

The decimal value of Khinchin's constant is given by:

${\displaystyle K_{0}=2.68545\,20010\,65306\,44530\dots }$ (sequence A002210 in the OEIS)

Although almost all numbers satisfy this property, it has not been proven for any real number not specifically constructed for the purpose. The following numbers whose continued fraction expansions apparently do have this property (based on empirical data) are:

• π
• Roots of equations with a degree > 2, e.g. cubic roots and quartic roots
• Natural logarithms, e.g. ln(2) and ln(3)
• The Euler-Mascheroni constant γ
• Apéry's constant ζ(3)
• The Feigenbaum constants δ and α
• Khinchin's constant

Among the numbers x whose continued fraction expansions are known not to have this property are:

• Rational numbers
• Roots of quadratic equations, e.g. the square roots of integers and the golden ratio ⁠${\displaystyle \varphi }$⁠ (however, the geometric mean of all coefficients for square roots of nonsquare integers from 2 to 24 is about 2.708, suggesting that quadratic roots collectively may give the Khinchin constant as a geometric mean);
• The base of the natural logarithm e.

Khinchin is sometimes spelled Khintchine (the French transliteration of Russian Хинчин) in older mathematical literature.