The Mean Number of Eigenvalues in $I_n$ When $a$ is Rational

In Section 3 of the complete project report, we show how to express the random variable $X_{n,a}$ as a sum of simpler random variables defined on the group of $n \times n$ permutation matrices. When $a$ is rational, it is possible to manipulate this sum in order to find the large $n$ limit of $E[X_{n,a}]$, yielding the formula in the first part of Theorem 1. In this section, we take a closer look at the limit $\lim_{n\rightarrow\infty}E[X_{n,a}]$, which is a function of $a$ and $\ell$. For convenience, we will denote it as $f(a,\ell)$; recall that

$$f\left({p\over q} , \ell\right) = \frac{1}{q} \ln \left(\frac{(q\ell)^{\lfloor q\ell\rfloor}} {\lfloor q\ell \rfloor !}\right).$$

The first thing to notice about the limiting function is that it depends only on the denominator $q$ of $a$. In the next section we explore how the function behaves as $q$ increases, but first we consider how it varies with $\ell$. The function $f(p/q, \ell)$ is shown in Figure 1 below, for several values of $q$.

Note that $f$ is zero if $\ell\le 1/q$. This follows from the fact that the eigenvalues occur only at rational points with denominators no greater than $n$, so the minimum distance from $p/q$ to any eigenvalue (other than those at $p/q$) is $1/qn$. As $\ell$ gets larger, each $f$ appears more and more like a straight line. It is possible to obtain an approximation of $f$ when the quantity $q\ell$ is large by using Stirling's formula ( $N!\approx N^N e^{-N}\sqrt{2\pi N}$) to approximate $\lfloor q\ell \rfloor !$. This shows that

$$f\left({p\over q} , \ell\right)\approx \ell - {1\over q}\ln\sqrt{2\pi q\ell}.$$

The approximating function for $q = 1$ is shown in Figure 2, which reveals that the approximation is quite good even for small values of $\ell$. The error in the approximation is graphed in Figure 3.

This approximation provides a comparison to the distribution of independent points. (Recall the discussion in Section 2.) The mean in that case was simply $\ell$, and we see that the mean number of eigenvalues for rational $a$ will always be less than $\ell$. It seems reasonable that the value of $f(p/q, \ell)$ should be less than the mean for the uniform case because there are a large number of eigenvalues at the endpoint $e^{2\pi i p/q}$ that are excluded from the interval $I_n$. The $O(\ln \ell)$ correction term may reflect the effect of this systematic exclusion, although it is not obvious that this term should increase as $\ln \ell$.