Lévy's constant

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In mathematics Lévy's constant (sometimes known as the Khinchin–Lévy constant) occurs in an expression for the asymptotic behaviour of the denominators of the convergents of continued fractions.[1] In 1935, the Soviet mathematician Aleksandr Khinchin showed[2] that the denominators qn of the convergents of the continued fraction expansions of almost all real numbers satisfy

Soon afterward, in 1936, the French mathematician Paul Lévy found[3] the explicit expression for the constant, namely

(sequence A086702 in the OEIS)

The term "Lévy's constant" is sometimes used to refer to (the logarithm of the above expression), which is approximately equal to 1.1865691104… The value derives from the asymptotic expectation of the logarithm of the ratio of successive denominators, using the Gauss-Kuzmin distribution. In particular, the ratio has the asymptotic density function[citation needed]

for and zero otherwise. This gives Lévy's constant as


The base-10 logarithm of Lévy's constant, which is approximately 0.51532041…, is half of the reciprocal of the limit in Lochs' theorem.



The proof assumes basic properties of continued fractions.

Let be the Gauss map.


where is the Fibonacci number.

Proof. Define the function . The quantity to estimate is then .

By the mean value theorem, for any ,

The denominator sequence satisfies a recurrence relation, and so it is at least as large as the Fibonacci sequence .

Ergodic argument[edit]

Since , and , we have

By the lemma,

where is finite, and is called the reciprocal Fibonacci constant.

By the Birkhoff's ergodic theorem, the limit converges to

almost surely, where is the Gauss distribution.

See also[edit]


  1. ^ A. Ya. Khinchin; Herbert Eagle (transl.) (1997), Continued fractions, Courier Dover Publications, p. 66, ISBN 978-0-486-69630-0
  2. ^ [Reference given in Dover book] "Zur metrischen Kettenbruchtheorie," Compositio Matlzematica, 3, No.2, 275–285 (1936).
  3. ^ [Reference given in Dover book] P. Levy, Théorie de l'addition des variables aléatoires, Paris, 1937, p. 320.
  4. ^ Ergodic Theory with Applications to Continued Fractions, UNCG Summer School in Computational Number Theory University of North Carolina Greensboro May 18 - 22, 2020. Lesson 9: Applications of ergodic theory

Further reading[edit]

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