Lévy's constant

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Numbers

normalisation

value

$e^\beta$:

3.275822918721811159787681882453843863608475525982374149405198924190723215644960355181277540479174529

comment: Lévy's constant, the almost-sure limit of $q_n^{1/n}$ [5].

$\beta$:

1.186569110415625452821722975947237120568356536472054335954254298652809632056254443300348301108486876

comment: The Khinchin-Lévy constant $\beta=\log(e^\beta)=\pi^2/(12\log 2)$, the almost-sure limit of $\frac{1}{n}\log q_n$ [6].

Definition

Let $p_n/q_n$ be the convergents of the regular continued fraction expansion of a real number $x$ [2]. For almost every $x$ the denominators grow at a rate that does not depend on $x$: $\lim_{n\to\infty}q_n^{1/n}=e^{\beta}$ with $\beta=\frac{\pi^2}{12\log 2}$ [3]. This table holds that limit in both of the normalisations in use: $e^\beta$, which is what is usually called Lévy's constant, and its logarithm $\beta$, which is called the Khinchin-Lévy constant.

Parameters

normalisation

— normalisation

Formulas

(1)

$\beta=\frac{\pi^2}{12\log 2}$ and $e^\beta=\exp\left(\frac{\pi^2}{12\log 2}\right)$, where $e$ is $e$ and $\pi$ is $\pi$.

Comments

(2)

Both rows are called a constant of Lévy in the literature, which is exactly why a database entry has to say which one it holds. The row $e^\beta$ is the limit of $q_n^{1/n}$ itself and is the one usually meant by "Lévy's constant" [3] [5]. The row $\beta$ is its logarithm, the almost-sure limit of $\frac{1}{n}\log q_n$, and is usually called the Khinchin-Lévy constant [6]. Lévy proved the limit in [1].

(4)

For the Gauss map $x\mapsto 1/x-\lfloor 1/x\rfloor$, which is what shifts a continued fraction expansion by one term [4], the Lyapunov exponent with respect to the invariant Gauss measure is $2\beta=\frac{\pi^2}{6\log 2}$.

Programs

(P1)

Sage

R = RealBallField(400)
beta = R.pi()^2 / (12 * R(2).log())
beta                                          # the Khinchin-Levy constant
beta.exp()                                    # Levy's constant

References

[1]

P. Lévy, Sur les lois de probabilité dont dépendent les quotients complets et incomplets d'une fraction continue, Bulletin de la Société Mathématique de France 57 (1929), 178-194. (doi)

Links

[2]

MathWorld: Regular Continued Fraction

[3]

MathWorld: Levy Constant

[4]

Wikipedia: Gauss map

[5]

OEIS A086702: decimal expansion of Levy's constant

[6]

OEIS A100199: decimal expansion of $\pi^2/(12\log 2)$

Similar tables

Khinchin's means $K_p$ — the other almost-sure limit of the regular continued fraction, taken over the partial quotients rather than over the denominators of the convergents

Lochs's constant — equals $\frac{\log 10}{2\beta}$, so it is this constant read as a rate of conversion between decimal digits and partial quotients

$\pi$ — appears in the closed form (1)

$e$ — the base of $e^\beta$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: yes (it holds the constant in both normalisations in use, $e^\beta$ and $\beta$)

How they were obtained:

Both rows follow from (1) and were computed in real ball arithmetic, with $\pi$ and $\log 2$ taken as balls rather than as rounded constants, so every stored digit is covered by the enclosure. Nothing here is a limit that had to be estimated: the closed form is a theorem, and what the computation has to be careful about is only the arithmetic.

The stored values were compared with the decimal expansions in OEIS A086702 and A100199 [5] [6], and $\log(e^\beta)=\beta$ was checked on them.