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 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 D_r denotes the discrepancy of the ball of radius r, then, for an absolute constant C>0 and for every set of points ${x_j}^N_{j=1}$, one has $\int_M|D_r(x)|^2d\mu(x)\ge N^{-1-1/d}$. The conditions on r that we have depend on the dimension d of the manifold and cannot be achieved when d≡1(mod 4). Nonetheless, we prove a weaker estimate for such dimensions as well.