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Khinchin proved that

For almost all reals $r$ with continued fraction representation $[a_o; a_1, a_2, \dots ]$ the sequence $K_n = \left(\prod_{i=1}^{n} a_i\right)^{1/n}$ converges to a constant $K$ (Khinchin's constant) independent of $r$.

This is quite surprising, and I was wondering if there is anything deeper going on behind the scenes. Is there something deeper? Is there any intuitive explanation of why this must be true?

  • The proof of the result nowadays seems to go through ergodicity of the Gauss/shift map on a measure equivalent to Lebesgue. It is mentioned in the comments that there is an "analogous result" for arithmetic mean of decimals in decimal expansion which feels a lot more intuitive. Perhaps an answer would rephrase or highlight a way to view the standard proof to make it "just as" intuitive. By the way, the analogous result doesn't feel like it should be evidence for Khinchin's result. For example, the numbers in decimal sequences are bounded and it basically says the digits are random while Khinchin's result seems to actually show that $a_n$ are actually controlled a lot, while still being unbounded in general. In fact, the arithmetic mean version for continued fractions diverge almost everywhere.

  • Another approach might be more geometric or topological. For example, continued fraction expansions can be seen as train track expansion sequences on the torus, and the real numbers as curves with specified slopes. Also, there is this nice looking paper The Modular Surface and Continued Fractions by Caroline Series which feel promising and might be on the track of giving a geometric/topological reason and interpretations for $K$ ($K$ isn't explicitly mentioned).

Arnaud Mortier
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    Khinchin's little book is translated and inexpensive, http://store.doverpublications.com/0486696308.html – Will Jagy Jul 09 '14 at 20:13
  • For something similar, see [this](http://math.stackexchange.com/questions/834220). – Lucian Jul 09 '14 at 22:46
  • Just a quote from http://en.wikipedia.org/wiki/Khinchin%27s_constant "... it can be proved that $T$ is an ergodic transformation of the measurable space $I$ endowed with the probability measure $\mu $ (this is the hard part of the proof). The ergodic theorem then says that for any $\mu $-integrable function $f$ on $I$, the average value of $f \left( T^{k} x \right)$ is the same for almost all $x$ ..." – rtybase May 06 '15 at 17:32
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    Michael Lugo has pointed out that there is an analogous theorem about the arithmetic mean of the decimal digits of a real number: For almost all reals $r$ with decimal digit representation $0.d_0d_1d_2\ldots$, the sequence $J_n = \frac1n\sum_{i=1}^n d_n$ converges to a constant $J$ independent of $r$. The value of $J$ is exactly $\frac92$. Seen in this light, the Khinchin constant seems much less mysterious. – MJD Oct 22 '15 at 17:07
  • @Mason Yes, I was hoping to add context which might make this question more interesting to people. I don't think it changes anything, just adds to it. Although I am considering rolling back and just asking a new one with this additional context –  Dec 01 '19 at 01:50
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    Also [here's](http://plouffe.fr/simon/Phys%20et%20Math/Continued%20Fractions%20Khinchin.pdf) Khinchin's text. – Mason Dec 01 '19 at 02:59
  • @Mason To be honest I had been thinking about just removing that second question, as it probably should have been a separate question anyways. I think it would be hard to describe and I don't think it is true the "sums of non random is non random." For example if you take a random $x$, then double all the digits in the continued fraction to get a new number $x'$ in the set of measure zero (non-random). So it a "large" measure zero set (I might conjecture as a generating set they could generate the reals). Also it is dense since rationals are not random. –  Dec 01 '19 at 03:14
  • Yah, that would be interesting @Mason. As stated originally it felt a little vague, but it could interesting and answerable with some more specifics in there (like your arithmetic question) –  Dec 01 '19 at 03:24
  • [Another related question](https://math.stackexchange.com/questions/3459713/is-khinchins-constant-really-about-real-numbers) whose answer might be relevant to this one. – Arnaud Mortier Dec 02 '19 at 15:56

2 Answers2

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Here is what I think of as the standard explanation. I learned this from The Art of Computer Programming, Section 4.5.3.

Start with some number $z$ whose continued fraction we want to find. I'll review the continued fraction algorithm: Let $a_1 = \lfloor z \rfloor$ and let $r_1 = x-a_1$. Then let $a_2 = \lfloor r_1^{-1} \rfloor$ and let $r_2 = r_1^{-1} - a_2$. Continue in this way to define $r_k$ and $a_k$. The $a_k$ are the coefficients of the continued fraction. Our first task will be to understand the $r_k$.

If $z$ is chosen randomly, then $r_1$ is equally likely to be anywhere in $[0,1)$. But the same is not true of $r_2$. We have $r_2 \in [0,x]$ if and only if $r_1 \in \bigcup_{k=1}^{\infty} [(k+x)^{-1}, k^{-1}]$, so the probability that $r_2$ is in $[0,x]$ is $\sum_{k=1}^{\infty} \left( \tfrac{1}{k} - \tfrac{1}{k+x} \right)$. More generally, let $F_n(x)$ be the probability that $r_n \in [0,x]$, then $$F_{n+1}(x) = \sum_{k=1}^{\infty} \left( F_n(\tfrac{1}{k}) - F_n(\tfrac{1}{k+x} ) \vphantom{\sum} \right).$$

If we imagine that $\lim_{n\to \infty} F_n(x)$ will exist, then the limiting function $F$ should obey $$F(x) = \sum_{k=1}^{\infty} \left( F(\tfrac{1}{k}) - F(\tfrac{1}{k+x} ) \vphantom{\sum} \right).$$

One function that obeys this property is $\log (1+x)$. Let's verify this: $$ \sum_{k=1}^{N} \left( \log \frac{k+1}{k} - \log \frac{k+x+1}{k+x} \right) \sum_{k=1}^{N} \left( \log (k+1) - \log k - \log (k+x+1) + \log (1+x) \right).$$ This telescope to $$\log (N+1) - \log 1 - \log (N+x+1) + \log(1+x) = \log \frac{N+1}{N+x+1} + \log (1+x).$$ Sending $N \to \infty$, we get $\log (1+x)$. This argument was neutral as to what the base of the logarithm was but, since we want $F(1)=1$, we should take log base two.

Of course, I am not going to prove that the limit exists or that it takes this value -- this is a hard theorem! But let's see why it implies the rest. The probability that $a_k = b$ is the probability that $r_{k-1} \in (\tfrac{1}{b+1}, \tfrac{1}{b}]$, or $F_{k-1}(\tfrac{1}{b}) - F_{k-1}(\tfrac{1}{b+1})$. If $F_k$ approaches $\log_2 (1+x)$, then the probability that $a_k$ is $b$ approaches $\log_2 \tfrac{b+1}{b} - \log_2 \tfrac{b+2}{b+1} = \log_2 \tfrac{(b+1)^2}{b(b+2)}$. Note that this is $\approx \tfrac{1}{b^2}$.

Intuitively then, if $g$ is a function from $\mathbb{Z}_{>0}$ to $\mathbb{R}$, we would expect the average value of $g(a_k)$ to be $\sum_{b=1}^{\infty} g(b) \log_2 \tfrac{(b+1)^2}{b(b+2)}$. It wouldn't be a good idea to take $g(x) = x$, since then the sum would diverge. But, if $g(x)=\log x$, then the sum converges to the log of the Khinchin constant, as one would hope for.

There is a subtle switch in the last paragraph. Before the last paragraph, $z$ was a random variable, and I looked at how the probability distribution of $a_k$ behaved as $k \to \infty$. In the last paragraph, and in Khinchin's theorem, I took a single $z$ and asked about the average behavior of $a_k$ for that $z$. This is a hard issue, but ergodic theory has standard tools to justify it.

David E Speyer
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  • I tried to make graphics to go with these answer and failed. If you think you can, check out https://mathematica.stackexchange.com/q/210784/1200 – David E Speyer Dec 05 '19 at 19:09
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If you want a geometric explanation of the result, the existence of Khinchin's constant is a consequence of the erodicity property of the geodesic flow on the modular surface (more precisely, on the modular orbifold).

See

Robert Hines, Simple continued fractions and the geodesic flow on the modular surface

for details: A direct proof of ergodicity is given on page 3 (takes about half a page). On page 4, the existence of Khinchin's constant is derived as a one-line calculation using a suitably chosen measurable function (it relies on Birkhoff's ergodic theorem).

@Paul Plummer's guess about train-tracks was a close miss: The theory of train tracks is a combinatorial tool to describe measured geodesic laminations on surfaces. The modular orbifold carries no geodesic laminations whatsoever, but if you drop the non-intersection property in the definition of geodesic laminations you arrive to the notion of geodesic currents, i.e. flow-and-flip-invariant measures on the unit tangent bundle. [Incidentally, afik, there is no known combinatorial tool, similar to train-tracks in the lamination case, which can be used to study general geodesic currents on hyperbolic surfaces. The continued fractions can be regarded as such a combinatorial tool in the very special case at hand.]

The Liouville measure is one such geodesic current. The ergodicity of the geodesic flow with respect to this measure (due to Hopf) gets you to the answer (the existence of Khinchin's constant).

A nice self-contained exposition of geodesic laminations and currents can be found in these lecture notes:

Javier Aramayona and Christopher J. Leininger, Hyperbolic structures on surfaces and geodesic currents.

Ergodicity of the geodesic flow with respect the to Liouville measure holds in much greater generality, say, for strictly negatively curved manifolds/orbifolds of finite volume (this is due to Hopf). There are many other nice connections of the ergodic theory on locally homogeneous spaces ("homogeneous dynamics") and the number theory, but that would be a separate question, more suitable for Mathoverflow...

Moishe Kohan
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