Curious Constants (part 3 of 4)

With our background on continued fractions behind us, let’s get back into some constants.

The Second Batch

0.10841… & 0.27394419… & 0.48267728…

I include more decimals here (than for previous constants in this talk), because without them, the interest is lost. While the discussion of continued fractions above only deals with simple continued fractions, I’ll take the decimals above as the values for non-simple continued fractions, and then notice that

\begin{array}{rcl} 0.10841 &\approx& \cfrac{1}{1+\cfrac{1}{0+\cfrac{1}{8+\cfrac{1}{4+\cfrac{1}{1+\ddots}}}}} \\ 0.27394419 &\approx& \cfrac{2}{7+\cfrac{3}{9+\cfrac{4}{4+\cfrac{1}{9+\ddots}}}} \\ 0.48267728 &\approx& \cfrac{4}{8+\cfrac{2}{6+\cfrac{7}{7+\cfrac{2}{8+\ddots}}}} \end{array}

These constants go by the name of their discoverer, M. Trott, who wrote about them just in the past decade. Existence and uniqueness of such constants is an open question.

In order to introduce some notation I’ll use in the next section, let me also write the second example above as \frac{2}{7+}\frac{3}{9+}\frac{4}{4+}\frac{1}{9+}.

0.596…

This has continued fraction expansion

0.596 \approx \frac{1}{1+}\frac{1}{1+}\frac{1}{1+}\frac{2}{1+}\frac{2}{1+}\frac{3}{1+}\frac{3}{1+}\frac{4}{1+}\frac{4}{1+}\cdots

That is, after an initial 1, all “numerators” repeate twice and then increment by 1. This expression is obtained from the function

u(x)=\frac{1}{1+}\frac{x}{1+}\frac{x}{1+}\frac{2x}{1+}\frac{2x}{1+}\frac{3x}{1+}\frac{3x}{1+}\frac{4x}{1+}\frac{4x}{1+}\cdots

by evaluating at x=1. So, where did this continued fraction come from? Euler came to this continued fraction as one for

u(x)=1-x+2x^2-6x^3+(4!)x^4-(5!)x^5+\cdots.

That is, u(1)=\sum_{n=0}^{\infty}(-1)^n(n!)\approx 0.596. Euler, in fact, determined this sum in 4 different ways, obtaining approximations near 0.59 each time. Another way he did this was to notice that

s(x)=xu(x)=x-x^2+2x^3-(3!)x^4+(4!)x^5-\cdots

satisfies (formally) the differential equation s'+\frac{s}{x^2}=\frac{1}{x}. Using integrating factors, this means that s(x)=e^{1/x}\int_0^x\frac{e^{-1/t}}{t}dt. Making the substitution v=e^{1-1/t} and then evaluating at 1, we get s(1)=1\cdot u(1)\approx 0.596=\int_0^1 \frac{dv}{1-\ln v}. The infinite sum of alternating factorials can be recovered from this integral by repeatedly applying integration by parts.

3.275… & 2.685

In the exposition on continued fractions we saw lower bounds on the denominators of convergents. It turns out an upper bound exists as well (at least, for the continued fractions of almost all \alpha). Even stronger, it can be shown that there is a constant which I will denote \gamma (following Khinchin, and not to be confused with the Euler-Mascheroni constant) such that for almost all \alpha, \sqrt[n]{q_n}\rightarrow \gamma. Levy showed that \ln(\gamma)=\frac{\pi^2}{12\ln 2}\approx 1.186, and so \gamma\approx 3.275.

Khinchin, in the mid 1930s, proved the following theorem. Suppose that f:\mathbf{N}\to \mathbf{R}_{\geq 0} and there are positive constants C,\delta such that f(r)<Cr^{\frac{1}{2}-\delta}. Then for almost all \alpha\in (0,1),

\displaystyle \lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^{n}f(a_k)=\sum_{r=1}^{\infty}f(r)\frac{\ln(1+1/(r(r+2)))}{\ln(2)}

Take, for example, f(r) to be 1 if r=k and 0 otherwise. Then the sum on the left above counts the number of occurences of k in the first n elements of \alpha, so the average determines the density of ks. The theorem then tells us that for almost any \alpha\in (0,1), the density of ks in the elements of \alpha is a positive constant d(k). For instance, d(1)=\frac{\ln(4)-\ln(3)}{\ln(2)}\approx 0.41503.

A more interesting example is found by taking f(r)=\ln(r). The statement of the theorem then gives that, almost everywhere,

\displaystyle\sqrt[n]{a_1\cdots a_n}\rightarrow \prod_{r=1}^{\infty}\left(1+\frac{1}{r(r+2)}\right)^{\frac{\ln r}{\ln 2}}\approx 2.685

Humorously, “almost everywhere” leaves out two of the examples of continued fractions you might first consider, the golden ratio \phi=[1;1,1,1,\ldots] and e=[2;1,2,1,1,4,1,1,6,1,1,8,\ldots].

Other posts in this series

  1. The First Batch
  2. Continued Fractions
  3. The Second Batch
  4. Resources
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5 Responses to “Curious Constants (part 3 of 4)”

  1. sumidiot Says:

    In the first posted version of this, I had Khinchin’s theorem incorrectly stated. The ratio was stated as 1/(r(r+1)) in the natural log on the right side of the equality, when it should have been 1/(r(r+2)). This has since been corrected (I hope!).

  2. sumidiot Says:

    Just before giving my talk, and after posting this series, I found another constant that I thought would be fun to talk about. It is from Lochs’ theorem, concerning the number of correct decimal digits you get from using however many terms of a contined fraction expansion. As you might guess, I found it on Wikipedia.

  3. Curious Constants (part 2 of 4) « Sumidiot’s Blog Says:

    […] Blog The math fork of sumidiot.blogspot.com? « Curious Constants (part 1 of 4) Curious Constants (part 3 of 4) […]

  4. Curious Constants (part 1 of 4) « Sumidiot’s Blog Says:

    […] The Second Batch […]

  5. Curious Constants (part 4 of 4) « Sumidiot’s Blog Says:

    […] Blog The math fork of sumidiot.blogspot.com? « Curious Constants (part 3 of 4) Piecewise Functions […]

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