Summary: We study the problem originally communicated by E. Meckes on the asymptotics for the eigenvalues of the kernel of the unitary eigenvalue process of a random \(n \times n\) matrix. The eigenvalues \(p_j\) of the kernel are, in turn, associated with the discrete prolate spheroidal wave functions. We consider the eigenvalue counting function \(|G(x, n)| := \#\{j: p_j > Ce^{-xn}\}\), (\(C>0\) here is a fixed constant) and establish the asymptotic behavior of its average over the interval \(x\in(\lambda - \varepsilon, \lambda + \varepsilon)\) by relating the function \(|G(x, n)|\) to the solution \(J(q)\) of the following energy problem on the unit circle \(S^1\), which is of independent interest. Namely, for given \(\theta\), \(0< \theta < 2\pi\), and given \(q\), \(0 < q < 1\), we determine the function \(J(q) = \inf\{I(\mu): \mu\in\mathcal{P}(S^1)\), \(\mu(A_\theta) = q\}\), where \(I(\mu) := \int\int\log\frac{1}{|z - \zeta|} d\mu(z)d\mu(\zeta)\) is the logarithmic energy of a probability measure \(\mu\) supported on the unit circle and \(A_\theta\) is the arc from \(e^{-i\theta/2}\) to \(e^{i\theta/2}\).