Lévy's constant
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.<ref> Template:Citation </ref> In 1935, the Soviet mathematician Aleksandr Khinchin showed<ref> [Reference given in Dover book] "Zur metrischen Kettenbruchtheorie," Compositio Matlzematica, 3, No.2, 275–285 (1936). </ref> that the denominators qn 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<ref> [Reference given in Dover book] P. Levy, Théorie de l'addition des variables aléatoires, Paris, 1937, p. 320. </ref> the explicit expression for the constant, namely
- <math>e^{\beta} = e^{\pi^2/(12\ln2)} = 3.275822918721811159787681882\ldots</math> (sequence A086702 in the OEIS)
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 functionTemplate:Citation needed
<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.
ProofEdit
<ref>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</ref>
The proof assumes basic properties of continued fractions.
Let <math>T : x \mapsto 1/x \mod 1</math> be the Gauss map.
LemmaEdit
<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 argumentEdit
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_n</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 alsoEdit
ReferencesEdit
<references/>
Further readingEdit
External linksEdit
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