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 simple continued fractions.
In 1935, the Soviet mathematician Aleksandr Khinchin showed that the denominators q<sub>n</sub> of the convergents of the continued fraction expansions of almost all real numbers satisfy
:<math>\lim_{n \to \infty}{q_n}^{1/n}= e^{\beta}</math>
Soon afterward, in 1936, the French mathematician Paul Lévy found the explicit expression for the constant, namely
:<math>e^{\beta} = e^{\pi^2/(12\ln2)} = 3.275822918721811159787681882\ldots</math>
The term "Lévy's constant" is sometimes used to refer to <math>\pi^2/(12\ln2)</math> (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
<math>f(z)=\frac{1}{z(z+1)\ln(2)}</math>
for <math>z \geq 1</math> and zero otherwise. This gives Lévy's constant as
<math>\beta=\int_1^\infty\frac{\ln z}{z(z+1)\ln 2}dz=\int_0^1\frac{\ln z^{-1{(z+1)\ln 2}dz=\frac{\pi^2}{12\ln 2}</math>.
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
The proof assumes basic properties of continued fractions.
Let <math>T : x \mapsto 1/x \mod 1</math> be the Gauss map.
Lemma
<math display="block">|\ln x - \ln p_n(x)/q_n(x)| \leq 1/q_n(x) \leq 1/F_n</math>where <math display="inline">F_n</math> is the Fibonacci number.
Proof. Define the function <math display="inline">f(t) = \ln\frac{p_n + p_{n-1}t}{q_n + q_{n-1}t}</math>. The quantity to estimate is then <math>|f(T^n x) - f(0)| </math>.
By the mean value theorem, for any <math display="inline">t\in [0, 1]</math>,<math display="block">
|f(t)-f(0)| \leq \max_{t \in [0, 1]}|f'(t)| = \max_{t \in [0, 1]} \frac{1}{(p_n + tp_{n-1})(q_n + tq_{n-1})} = \frac{1}{p_nq_n} \leq \frac{1}{q_n}
</math>The denominator sequence <math>q_{0}, q_1, q_2, \dots</math> satisfies a recurrence relation, and so it is at least as large as the Fibonacci sequence <math>1, 1, 2, \dots</math>.
Ergodic argument
Since <math display="inline">p_n(x) = q_{n-1}(Tx)</math>, and <math display="inline">p_1 = 1</math>, we have<math display="block">-\ln q_n = \ln\frac{p_n(x)}{q_n(x)} + \ln\frac{p_{n-1}(Tx)}{q_{n-1}(Tx)} + \dots + \ln\frac{p_1(T^{n-1}x)}{q_1(T^{n-1} x)}</math>By the lemma,
<math display="block">
-\ln q_n = \ln x + \ln Tx + \dots + \ln T^{n-1}x + \delta
</math>
where <math display="inline">|\delta| \leq \sum_{k=1}^\infty 1/F_k</math> is finite, and is called the reciprocal Fibonacci constant.
By Birkhoff's ergodic theorem, the limit <math display="inline">\lim_{n \to \infty}\frac{\ln q_n}{n}</math> converges to<math display="block">
\int_0^1 ( -\ln t )\rho(t) dt = \frac{\pi^2}{12\ln 2} </math> almost surely, where <math>\rho(t) = \frac{1}{(1+t) \ln 2}</math> is the Gauss distribution.
See also
- Khinchin's constant