Single radius spherical cap discrepancy on compact two-point homogeneous spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913380110958592 |
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| author | Brandolini, Luca Gariboldi, Bianca Gigante, Giacomo Monguzzi, Alessandro |
| author_facet | Brandolini, Luca Gariboldi, Bianca Gigante, Giacomo Monguzzi, Alessandro |
| contents | 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 $ρ$ so that $(\mathcal M, ρ)$ is a two-point homogeneous space and with the Riemannian measure $μ$, 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 $\int_{\mathcal M} |D_{r}(x)|^2\, dμ(x)\geqslant C N^{-1-\frac1d}$. The conditions on $r$ that we have depend on the dimension $d$ of the manifold and cannot be achieved when $d \equiv 1 \ ( \operatorname{mod}4)$. Nonetheless, we prove a weaker estimate for such dimensions as well. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_03830 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Single radius spherical cap discrepancy on compact two-point homogeneous spaces Brandolini, Luca Gariboldi, Bianca Gigante, Giacomo Monguzzi, Alessandro Classical Analysis and ODEs Number Theory 11K38, 43A85, 33C45 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 $ρ$ so that $(\mathcal M, ρ)$ is a two-point homogeneous space and with the Riemannian measure $μ$, 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 $\int_{\mathcal M} |D_{r}(x)|^2\, dμ(x)\geqslant C N^{-1-\frac1d}$. The conditions on $r$ that we have depend on the dimension $d$ of the manifold and cannot be achieved when $d \equiv 1 \ ( \operatorname{mod}4)$. Nonetheless, we prove a weaker estimate for such dimensions as well. |
| title | Single radius spherical cap discrepancy on compact two-point homogeneous spaces |
| topic | Classical Analysis and ODEs Number Theory 11K38, 43A85, 33C45 |
| url | https://arxiv.org/abs/2406.03830 |