Random Points on S³ with Small Logarithmic Energy

We analyse several constructions of random point sets on the sphere S³⁴ evaluating and comparing them through their discrete logarithmic energy: {equation*} E_0(ω_N) = ∑_{{i, j=1\\ i ≠ j}}N log1/\|x_i - x_j\|, \; where\; ω_N=\{x_1,…,x_N\} ⊂ S^3. {equation*} Using the Hopf fibration, we lift a range of well-distributed families of points from the $2$-dimensional sphere - including uniformly random points, antipodally symmetric sets, determinantal point processes, and the Diamond ensemble - to S³, in order to assess their energy performance. In particular, we carry out this asymptotic analysis for the Spherical ensemble (a well known determinantal point process on S²), obtaining as a result a family of points on the $3$-dimensional sphere whose logarithmic energy is asymptotically the lowest achieved to date. This, in turn, provides a new upper bound for the minimal logarithmic energy on S³. Although an analytic treatment of the lifted Diamond ensemble remains elusive, extensive simulations presented here show that its empirical energies lie below all other deterministic and non-deterministic constructions considered. Together, these results sharpen the quantitative link between potential-theoretic optima on S² and S³ and provide both theoretical and numerical benchmarks for future work.