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 \(\mathcal {M}\) $\mathcal{M}$ endowed with a distance \(\rho \) $\rho$ so that \((\mathcal {M}, \rho )\) $(\mathcal{M},\rho )$ is a two-point homogeneous space and with the Riemannian measure \(\mu \) $\mu$ , we provide conditions on r such that if \(D_r\) $D_r$ denotes the discrepancy of the ball of radius r, then, for an absolute constant \(C>0\) $C>0$ and for every set of points \(\{x_j\}_{j=1}^N\) ${\{x_j\}}_{j=1}^N$ , one has \(\int _{\mathcal {M}} |D_{r}(x)|^2\, d\mu (x)\geqslant C N^{-1-\frac{1}{d}}\) ${\int }_{\mathcal{M}}{|D_r(x)|}^{2}d\mu (x)⩾C{N}^{-1-\frac{1}{d}}$ . The conditions on r that we have depend on the dimension d of the manifold and cannot be achieved when \(d \equiv 1 \ ( {\text {mod}}4)\) $d\equiv 1(\text{mod}4)$ . Nonetheless, we prove a weaker estimate for such dimensions as well.