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 $|G(x,n)|:=\#\{j:p_j>Ce^{-x n}\}$, ($C>0$ here is a fixed constant) and establish the asymptotic behavior of its average over the interval $x \in (λ-\varepsilon, λ+\varepsilon)$ by relating the function $|G(x,n)|$ to the solution $J(y)$ of the following energy problem on the unit circle $S^{1}$, which is of independent interest. Namely, for given $θ$, $0<θ< 2 π$, and given $q$, $0<q<1$, we determine the function $J(q) =\inf \{I(μ): μ\in \mathcal{P}(S^{1}), μ(A_θ) = q\}$, where $I(μ):= \iint \log\frac{1}{|z - ζ|} dμ(z) dμ(ζ)$ is the logarithmic energy of a probability measure $μ$ supported on the unit circle and $A_θ$ is the arc from $e^{-i θ/2}$ to $e^{i θ/2}$.