Irregularities of distribution on two point homogeneous spaces

Abstract

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 $\left(\{a_j{\}}_{j=1}^N,\{x_j{\}}_{j=1}^N\right)$ be a system of positive weights and points on $\mathcal{M}$ and let \[ D_{r}( x) =\sum_{j=1}^{N}a_{j}χ_{B_{r}(x)}(x_{j})-μ(B_{r}(x)) \] be the discrepancy associated with the ball $B_r(x)$. Then, if $d\not\equiv 1(\mathrm{mod}4)$, for any radius $0<r<\pi /2$, we obtain the sharp estimate \[ \int_{\mathcal{M}}\left( \left\vert D_{r}( x) \right\vert ^{2}+\left\vert D_{2r}( x) \right\vert ^{2}\right) dμ( x) \geqslant cN^{-1-\frac{1}{d}}. \]

20 pages