Failure of Orthogonality of Rounded Fourier Bases

The purpose of this note is to prove estimates for $$\mid \sum _{k=1}^{n}sign(\cos \left(\frac{2\pi a}{n}k\right))sign(\cos \left(\frac{2\pi b}{n}k\right))\mid ,$$ when $n$ is prime and $a,b\in \mathbb{N}$. We show that the expression can only be large if $a^{-1}b\in {\mathbb{F}}_n$ (or a small multiple thereof) is close to $1$. This explains some of the surprising line patterns in $A^{T}A$ when $A\in {\mathbb{R}}^{n\times n}$ is the signed discrete cosine transform. Similar results seem to exist at a great level of generality.