The Mean When $a$ is Irrational

The idea behind the second part of Theorem 1 is that we can approximate an irrational number by a sequence of rational numbers with increasing denominators. Thus, it might be reasonable to take the limit as $q$ approaches infinity of the function $f(p/q, \ell)$ in order to find the limit of $E[X_{n,a}]$ for irrational $a$. Figure 4 shows the behavior of $f(p/q, \ell)$ as $q$ increases.

In Figure 4, the functions $f(p/q, \ell)$ appear to be approaching the value of $\ell$ as $q$ gets larger. Using Stirling's formula as in the previous section shows that the limit as $q$ approaches infinity of $f(p/q, \ell)$ is in fact $\ell$. This hints at the connection between the two parts of Theorem 1. However, the proof of Part 2 of the theorem is not nearly as simple as this argument suggests, and in fact it does not apply to every irrational number (hence the phrase "almost every" in the statement of the theorem). The problem is that different irrational numbers allow different degrees of accuracy when we approximate them by rationals. One way to study such approximations is with the theory of continued fractions, and the proof of Part 2 of Theorem 1 is based on results that can be found in the book Continued Fractions by Khinchin [3]. (Also see the projects by Justin Miller [Families of Continued Fractions] or Santiago Canez [Continued Fraction Factorization] for more information about continued fractions.) Following is a brief outline of the content the proof.

The overall idea is to find a sequence of rational numbers $r_n/s_n$ converging to the irrational number $a$, and then to use the random variable $X_{n,\frac{ r_n}{s_n}}$ to approximate the value of $X_{n,a}$. If this approximation gets better as $n$ increases, we can hope to show that

$$\lim_{n \rightarrow \infty} \left\vert E \left[X_{n, a}\right] -E\left[ X_{n,\frac{r_n}{s_n}}\right]\right\vert=0.$$ (1)

This would imply that the limiting value of $E \left[X_{n, a}\right]$ will equal the limiting value of $E\left[X_{n,\frac{ r_n}{s_n}}\right]$, assuming this second limit exists. Since the sequence of denominators $s_n$ must increase without bound, the observations at the beginning of this section lead us to believe that the limiting value should be $\ell$. However, proving this requires showing that

$$\lim_{n \rightarrow \infty} \left\vert E\left[ X_{n,\frac{r_n}{s_n}}\right] -f\left(\frac{r_n}{s_n},\ell\right)\right\vert=0.$$ (2)

for the specific sequence $r_n/s_n$ that we chose.

We first consider the limit in Equation (2). This is where the error bound in Theorem 2 comes in. In order for the difference in Equation (2) to go to zero, the denominators $s_n$ cannot increase too fast relative to $n$. Specifically, according to Theorem 2 we must choose $r_n/s_n$ so that as $n$ approaches infinity, we have $s_n\rightarrow\infty$ but $(s_n\ln s_n)/n \rightarrow 0$. In this case, it follows that

$$\lim_{n\rightarrow\infty} E\left[X_{n,{r_n\over s_n}}\right] = \lim_{s_n\rightarrow\infty} f\left({r_n\over s_n},\ell\right) = \ell$$

as we had hoped.

The discussion in the previous paragraph shows that $s_n$ must be chosen to grow fairly slowly. At the same time, we must show that the limit in Equation (1) holds for the chosen sequence. In general, it is not clear that this limit should hold for any sequence, and the restrictions placed on our possible choices make it even less likely to hold. To establish this limit, we must carefully consider the difference between the values of $E[X_{n,a}]$ and $E\left[X_{n,{r_n\over s_n}}\right]$. When we shift the interval $I_n$ (with left endpoint $a$) to a nearby interval with left endpoint ${r_n/ s_n}$, any eigenvalues that occur at rational points between the left or right endpoints of the two intervals would be included in one interval but not the other. Hence, the difference between $E[X_{n,a}]$ and $E\left[X_{n,{r_n\over s_n}}\right]$ will be the expected number of eigenvalues in these two spaces where the intervals don't overlap. We can conclude that $E[X_{n,a}]\rightarrow\ell$ only if the expected number of these eigenvalues goes to zero.

Luckily, the theory of continued fractions allowed us to obtain an upper bound on the contribution of these eigenvalues, at least in some cases. Continued fractions provide a method to find the best rational approximations, or convergents, of an irrational number, and this procedure yields many results about the general problem of approximating irrational numbers. It turns out that the most "well-behaved" irrationals are often those that are poorly approximated, in the sense that it takes rational numbers with relatively large denominators to get close to the irrational point. Indeed, the crucial element in proving the second part of Theorem 1 was to restrict our attention to irrational numbers that did not allow too high a degree of approximation. For the specific condition on the irrational numbers to which the proof applies, see Theorem 8.1 in the complete paper (p. 26). For the class of irrational numbers described in that theorem, it is possible to construct a sequence $(r_n/s_n)$ which satisfies both conditions (1) and (2) above.

It is important to note that the proof given in the paper does not work for all irrational numbers. In fact, it can be shown that if $a$ is an irrational number that can be approximated extremely well, then $E[X_{n,a}]$ has an unbounded subsequence, so the mean cannot possibly converge to $\ell$ or any other finite value. In other words, it is possible to construct an irrational number $a$ for which the conclusion of Part 2 of Theorem 1 does not hold. The reason that we can claim that $E[X_{n,a}]\rightarrow\ell$ for "almost every" irrational number is that most irrationals do not allow a very high degree of approximation.