Equidistribution and separation characterization of logarithmic-energy optimality

Let $D(\omega_N)$ denote the spherical cap discrepancy of a sequence of configurations $\{\omega_N\}_{N=2}^{\infty}\subset\mathbb{S}^2$. Call the sequence well-separated when its points have separation bounded below on the natural $N^{-1/2}$ scale, and call it asymptotically optimal with respect to logarithmic energy when its logarithmic energy has the asymptotic minimal value to the relevant order. Equidistribution–optimality conjecture. The condition

$$\lim_{N\to\infty}D(\omega_N)=0$$

can replace the stronger discrepancy condition in the cited partial converse. Thus a sequence of configurations is equidistributed and well-separated if and only if it is asymptotically optimal with respect to logarithmic energy. The source presents this as a proposed relaxation of a known sufficient condition; the precise definition of well-separatedness and the exact normalization of asymptotic optimality are supplied earlier in the paper.