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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 $$\displaystyle D_{r}( x) =\sum _{j=1}^{N}a_{j}\chi _{B_{r}(x)}(x_{j})-\mu (B_{r}(x)) $$ be the discrepancy associated with the ball $$B_{r}( x) $$ . Then, if $$d\not \equiv 1(\operatorname {mod}4)$$ , for any radius $$0

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Irregularities of Distribution on Two-Point Homogeneous Spaces

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  • 发表日期：

    2025年

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  • Luca Brandolini,
  • Bianca Gariboldi,
  • Giacomo Gigante

摘要

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 $$\displaystyle D_{r}( x) =\sum _{j=1}^{N}a_{j}\chi _{B_{r}(x)}(x_{j})-\mu (B_{r}(x)) $$ be the discrepancy associated with the ball $$B_{r}( x) $$ . Then, if $$d\not \equiv 1(\operatorname {mod}4)$$ , for any radius $$0