Single radius spherical cap discrepancy on compact two-point homogeneous spaces

In this note we study estimates from below of the single radius spherical discrepancy in the setting of compact two-point homogeneous spaces. Namely, given a d-dimensional manifold M endowed with a distance ρ so that ( M, ρ) is a two-point homogeneous space and with the Riemannian measure μ, we provide conditions on r such that if Dᵣ denotes the discrepancy of the ball of radius r, then, for an absolute constant $C>0$ and for every set of points ⱼ\ⱼ₌₁N, one has ∫M |Dᵣ(x)|²\, dμ(x) C N⁻¹⁻¹ᵈ. The conditions on r that we have depend on the dimension d of the manifold and cannot be achieved when d ≡ 1 \ ( mod4). Nonetheless, we prove a weaker estimate for such dimensions as well.