Metric uniform distribution on analytic curves
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912576627015680 |
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| author | Bergelson, Vitaly Moreira, Joel |
| author_facet | Bergelson, Vitaly Moreira, Joel |
| contents | We obtain multidimensional metric uniform distribution results involving sequences in ${\mathbb R}^k$ parametrized by analytic curves. Our theorems extend the classical theorems of Weyl and Koksma in a variety of ways.
One of our main results implies that for any injective sequences $a_1,\dots,a_k:{\mathbb N}\to{\mathbb Z}$ the set
$$\Big\{(x_1,\dots,x_k)\in{\mathbb R}^k:\big(a_1(n)x_1,\dots,a_k(n)x_k\big)_{n\in{\mathbb N}}\text{ is uniformly distributed in }{\mathbb T}^k\Big\}$$
has full Lebesgue measure inside any non-degenerate analytic curve $γ\subset{\mathbb R}^k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06909 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Metric uniform distribution on analytic curves Bergelson, Vitaly Moreira, Joel Classical Analysis and ODEs We obtain multidimensional metric uniform distribution results involving sequences in ${\mathbb R}^k$ parametrized by analytic curves. Our theorems extend the classical theorems of Weyl and Koksma in a variety of ways. One of our main results implies that for any injective sequences $a_1,\dots,a_k:{\mathbb N}\to{\mathbb Z}$ the set $$\Big\{(x_1,\dots,x_k)\in{\mathbb R}^k:\big(a_1(n)x_1,\dots,a_k(n)x_k\big)_{n\in{\mathbb N}}\text{ is uniformly distributed in }{\mathbb T}^k\Big\}$$ has full Lebesgue measure inside any non-degenerate analytic curve $γ\subset{\mathbb R}^k$. |
| title | Metric uniform distribution on analytic curves |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2509.06909 |