Lévy's constant explained

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.In 1935, the Soviet mathematician Aleksandr Khinchin showed^[1] that the denominators q_n of the convergents of the continued fraction expansions of almost all real numbers satisfy

\lim _{n} {q ^{1 /n} _{n}} = e ^{\beta} Soon afterward, in 1936, the French mathematician Paul Lévy found^[2] the explicit expression for the constant, namelye ^{\beta} = e ^{\pi ^{2/(12ln2)}} = 3 . 275822918721811159787681882 \ldots

The term "Lévy's constant" is sometimes used to refer to

\pi ^{2/(12ln2)} (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 functionf(z) = (1)/(z(z +1 ) ln ( 2 ))

for

z \geq 1 and zero otherwise. This gives Lévy's constant as\beta = \int ^{inf ty (ln z)/(z(z +1 ) ln2)} _{1} dz = \int ^{1 (ln z ^{-1})/((z +1 ) ln2)} _{0} dz = (\pi ^{2})/(12ln2) .

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

Proof

^[3]

The proof assumes basic properties of continued fractions.

Let

T : x \mapsto 1 /x \mod 1 be the Gauss map.

Lemma

$$|\backslash ln\; x\; -\; \backslash ln\; p\_n(x)/q\_n(x)|\; \backslash leq\; 1/q\_n(x)\; \backslash leq\; 1/F\_n$$where $F\_n$ is the Fibonacci number.

Proof. Define the function $f(t)\; =\; \backslash ln\backslash frac$. The quantity to estimate is then

|f(T ^{n} x) - f( 0 )| .

By the mean value theorem, for any $t\backslash in\; [0,\; 1]$,$$|f(t)-f(0)|\; \backslash leq\; \backslash max\_|f\text{'}(t)|\; =\; \backslash max\_\; \backslash frac\; =\; \backslash frac\; \backslash leq\; \backslash frac$$The denominator sequence

q _{0} , q _{1,} q _{2,} ... satisfies a recurrence relation, and so it is at least as large as the Fibonacci sequence 1 , 1 , 2 , ... .

Ergodic argument

Since $p\_n(x)\; =\; q\_(Tx)$, and $p\_1\; =\; 1$, we have$$-\backslash ln\; q\_n\; =\; \backslash ln\backslash frac\; +\; \backslash ln\backslash frac\; +\; \backslash dots\; +\; \backslash ln\backslash frac$$By the lemma,$$-\backslash ln\; q\_n\; =\; \backslash ln\; x\; +\; \backslash ln\; Tx\; +\; \backslash dots\; +\; \backslash ln\; T^x\; +\; \backslash delta$$

where $|\backslash delta|\; \backslash leq\; \backslash sum\_^\backslash infty\; 1/F\_n$ is finite, and is called the reciprocal Fibonacci constant.

By the Birkhoff's ergodic theorem, the limit $\backslash lim\_\backslash frac$ converges to$$\backslash int\_0^1\; (-\backslash ln\; t)\backslash rho(t)\; dt\; =\; \backslash frac$$ almost surely, where

\rho(t) = (1)/(( 1+ t) ln2) is the Gauss distribution.

See also

• Khinchin's constant

References

1. Reference given in Dover book
2. Reference given in Dover book
3. 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

• Book: Khinchin, A. Ya.. Continued Fractions. 14 May 1997. Dover. 0-486-69630-8.