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The Mean When a Is Rational

Now we no longer assume a to be 0 and return to equation (10) for Xn,a:

 \begin{displaymath}X_{n,a}= \sum_{k=1}^n C_k \left(\left\lfloor ka + \frac{kl}{n} \right\rfloor - \lfloor ka \rfloor
\right).
\end{displaymath} (38)

A little thought shows that in general,

 \begin{displaymath}\lfloor x + y \rfloor = \cases{ \lfloor x \rfloor + \lfloor y...
...r
\lfloor x \rfloor + \lfloor y \rfloor + 1 & otherwise, \cr}
\end{displaymath} (39)

for any real numbers x and y. Applying (39) to the first term in (38) gives

\begin{displaymath}\left\lfloor ka + {kl \over n} \right\rfloor =
\cases{ \lflo...
...or + \left\lfloor {kl / n} \right\rfloor + 1 & otherwise, \cr}
\end{displaymath} (40)

and so
Xn,a = $\displaystyle \sum_{k=1}^n C_k \left(\left\lfloor ka + \frac{kl}{n} \right\rfloor -
\lfloor ka \rfloor \right)$ (41)
  = $\displaystyle \sum_{k=1}^n C_k \left\lfloor \frac{kl}{n} \right\rfloor + \sum_{...
...on\: \{ka\} +
\left\{ \frac{kl}{n} \right\} \geq 1, \atop 1 \leq k \leq n} C_k.$ (42)

Thus, the number of eigenvalues Xn,a in the interval $\left(e^{2 \pi i a},\,e^{2 \pi
i(a + l/n)}\right]$



2000-09-25