3.1 Sums of cosines at rotated roots

The discrete orthogonality of roots of unity manifests geometrically: the projections of equally-spaced points on the circle always balance around the origin.

Lemma 3.1.1

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For any \(N \geq 2\) and any \(\theta \in \mathbb {R}\),

\begin{equation} \sum _{k=0}^{N-1} \cos \left(\theta + \frac{2\pi k}{N}\right) = 0. \end{equation}

(See Lemma 3.1 in ../paper/chebyshev_circles.pdfthe paper)

Proof ▶

We use Euler’s formula: \(\cos \theta = \Re (\mathrm{e}^{i\theta })\). Thus

\begin{equation} \sum _{k=0}^{N-1} \cos \left(\theta + \frac{2\pi k}{N}\right) = \Re \left(\sum _{k=0}^{N-1} \mathrm{e}^{i(\theta + 2\pi k/N)}\right) = \Re \left(\mathrm{e}^{i\theta } \sum _{k=0}^{N-1} \mathrm{e}^{2\pi i k/N}\right). \end{equation}

By Lemma 2.2.1 with \(m = 1\) (noting that \(N \nmid 1\) for \(N \geq 2\)), the inner sum vanishes. Hence the entire expression is zero.

Lemma 3.1.2

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For any integers \(N \geq 1\), \(m\) with \(0 {\lt} m {\lt} N\), and any \(\theta \in \mathbb {R}\),

\begin{equation} \sum _{k=0}^{N-1} \cos \left(m\left(\theta + \frac{2\pi k}{N}\right)\right) = 0. \end{equation}

(See Lemma 3.2 in ../paper/chebyshev_circles.pdfthe paper)

Proof ▶

Using Euler’s formula, \(\cos (m\theta ) = \Re (\mathrm{e}^{im\theta })\), we have

\begin{equation} \sum _{k=0}^{N-1} \cos \left(m\left(\theta + \frac{2\pi k}{N}\right)\right) = \Re \left(\mathrm{e}^{im\theta } \sum _{k=0}^{N-1} \mathrm{e}^{2\pi i mk/N}\right). \end{equation}

Since \(0 {\lt} m {\lt} N\), we have \(N \nmid m\), so the inner sum vanishes by Lemma 2.2.1.