Largest eigenvalue of matrix associated to $\cos$ kernel

Question

Consider the kernel $K(x,y):=\cos(2\pi(x-y))$. Since $\cos(2\pi z)$ is the Fourier transform of a positive measure, it is positive definite (by Bochner's theorem) in the sense that for any $N\in\mathbb{N}$ and any distinct points $x_1,\ldots,x_N$, the symmetric matrix $\mathbf{A}=(K(x_i,x_j))_{i,j=1}^N$ is positive semidefinite. I am interested in an estimate for the difference between the largest and smallest eigenvalues of the matrix $\mathbf{A}$ as $N\rightarrow\infty$ and for general distinct points $x_1,\ldots,x_N$. Since all the eigenvalues are nonnegative the diagonals $K(x_i,x_i)=1$, I know that a crude upper for the largest eigenvalue is $\mathrm{Trace}(\mathbf{A})=N$. I also know that for general $N$, $\mathbf{A}$ is not strictly positive definite (for instance, see this paper), so that zero is an eigenvalue. But I am otherwise at a loss.