Irregularities of distribution on two point homogeneous spaces

We study the irregularities of distribution on two-point homogeneous spaces. Our main result is the following: let $d$ be the real dimension of a two point homogeneous space $\mathcal{M}$, let $(\{a_j{\}}_{j=1}^N,\{x_j{\}}_{j=1}^N)$ be a system of positive weights and points on $\mathcal{M}$ and let $$D_r(x)=\sum _{j=1}^N{a}_j{\chi }_{B_r(x)}(x_j)-\mu (B_r(x))$$ be the discrepancy associated with the ball $B_r(x)$. Then, if $d\equiv ̸1(\mathrm{mod}4)$, for any radius $0<r<\pi \mathrm{/}2$, we obtain the sharp estimate $${\int }_{\mathcal{M}}({\mid D_r(x)\mid }^2+{\mid D_{2r}(x)\mid }^2)d\mu (x)⩾c{N}^{-1-\frac{1}{d}}\mathrm{.}$$