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Conclusion

At this point we can ask how the result in Theorem 1 compares with the distribution of random independent points. Recall that the mean in that case was just l. Figure 2 shows another graph of f vs. l for different values of q, this time plotted over a wider range of l values. Notice that each f appears more and more like a straight line as l increases.

Using Stirling's formula as before, we can obtain an approximation of $f\left({p \over q},l\right)$ for large l values. Expanding the first and third terms in (66) as Taylor series reveals that f can be approximated by $l - (\ln l)/2q + O(1)$. Although it is not obvious that f should have this form, the negative sign for the correction term does make sense. Even though the eigenvalues become more evenly distributed as n increases, there are a large number of them at the endpoint $e^{2\pi ip/q}$ that are always excluded from the interval In. The correction term may reflect the effect of this exclusion.

Remark. The results derived in this paper are for half-open intervals, rather than for more standard open or closed intervals. This was done mainly to simplify notation as much as possible. In fact, the limit for an open interval of the form


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2000-09-25