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

Abstract

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 $\mathcal{M}$ endowed with a distance $\ddot{I}$ so that $(\mathcal{M},\ddot{I})$ is a two-point homogeneous space and with the Riemannian measure $\hat{I}¼$, 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 $\{x_j{\}}_{j=1}^N$, one has ${\int }_{\mathcal{M}}|D_r(x){|}^{2}d\hat{I}¼(x)⩾C{N}^{-1-\frac{1}{d}}$. The conditions on $r$ that we have depend on the dimension $d$ of the manifold and cannot be achieved when $d\equiv 1(\mathrm{mod}4)$. Nonetheless, we prove a weaker estimate for such dimensions as well.

16 pages