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This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point potentials. In particular, we study three-point analogues of the sphere packing problem and the optimization problem for $p$-frame energies based on three points. It turns out that both problems are inherently connected to the problem of nearly orthogonal sets by Erdős. As the outcome, we provide a new solution of the Erdős problem from the three-point packing perspective. We also show that the orthogonal basis uniquely minimizes the $p$-frame three-point energy when $0>p>1$in all dimensions. The arguments make use of multivariate polynomials employed in semidefinite programming and based on the classical Gegenbauer polynomials. For $p=1$, we completely solve the analogous problem on the circle. As for higher dimensions, we show that the Hausdorff dimension of minimizers is not greater than $d-<\#comment/>2$for measures on ${\mathbb{S}}^{d-<\#comment/>1}$. 

Abstract Determinantal point processes exhibit an inherent repulsive behavior, thus providing examples of very evenly distributed point sets on manifolds. In this paper, we study the so‐called harmonic ensemble, defined in terms of Laplace eigenfunctions on the sphere and the flat torus , and the so‐called spherical ensemble on , which originates in random matrix theory. We extend results of Beltrán, Marzo, and Ortega‐Cerdà on the Riesz ‐energy of the harmonic ensemble to the nonsingular regime , and as a corollary find the expected value of the spherical cap discrepancy via the Stolarsky invariance principle. We find the expected value of the discrepancy with respect to axis‐parallel boxes and Euclidean balls of the harmonic ensemble on . We also show that the spherical ensemble and the harmonic ensemble on and with points attain the optimal rate in expectation in the Wasserstein metric , in contrast to independent and identically distributed random points, which are known to lose a factor of .