Lower and upper bounds for configurations of points on a sphere

We present a new proof (based on spectral decomposition) of a bound originally proved by Sidelnikov, Math. USSR-Sb. 24, 147 (1974), for the frame potentials ∑ijPi⋅Pjℓ on a unit–sphere in d dimensions. Sidelnikov’s bound is a special case of the lower bound for the weighted sums ∑ijfifjPi⋅Pjℓ, where fi > 0 are scalar quantities associated to each point on the sphere, which we also prove using spectral decomposition. Moreover, in three dimensions, again using spectral decomposition, we find a sharp upper bound for ∑ijkNPi×Pj⋅Pk2. We explore two applications of these bounds: first, we examine configurations of points corresponding to the local minima of the Thomson problem for N = 972; second, we analyze various distributions of points within a three-dimensional volume, where a suitable weighted sum is defined to satisfy a specific bound.