论文标题
两个点均匀空间上的分布不规则
Irregularities of distribution on two point homogeneous spaces
论文作者
论文摘要
我们研究了两点均匀空间上分布的不规则性。我们的主要结果是以下结果：让$ d $是两个均质空间$ \ MATHCAL {m} $的真正维度点$ \ MATHCAL {M} $，让\ [d_ {r}（x）= \ sum_ {j = 1}^{n} a_ {j}χ_{b_ {b_ {r}（x）}（x_ {j}）（x_ {j}） - μ（x_ {j}） - μ（b_ v {b_ {r} = $ necipts we comptional wes ne companie x）$。然后，如果$ d \ not \ equiv 1（\ operatorName {mod} 4）$，对于任何半径$ 0 <r <π/2 $，我们获得了敏锐的估计\ [\ int _ {\ intcal {\ mathcal {m}}}} \ left（\ left（\ left） d_ {2r}（x）\ right \ vert ^{2} \ right）dμ（x）\ geqslant cn ^{ - 1- \ frac {1} {d}}}。 \]
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 $d\not \equiv 1(\operatorname{mod}4)$, for any radius $0<r<π/2$, we obtain the sharp estimate \[ \int_{\mathcal{M}}\left( \left\vert D_{r}( x) \right\vert ^{2}+\left\vert D_{2r}( x) \right\vert ^{2}\right) dμ( x) \geqslant cN^{-1-\frac{1}{d}}. \]
