A generalization of van der Corput's Difference Theorem
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866913468197634048 |
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| author | Farhangi, Sohail |
| author_facet | Farhangi, Sohail |
| contents | We prove a generalization of van der Corput's Difference Theorem in the theory of uniform distribution by establishing a connection with unitary operators that have Lebesgue spectrum. This allows us to show, for example, that if $(x_n)_{n = 1}^\infty \subseteq [0,1]$ is such that $(x_{n+h}-x_n)_{n = 1}^\infty$ is uniformly distributed for all $h \in \mathbb{N}$, then $(x_{n_k})_{k = 1}^\infty$ is uniformly distributed, where $(n_k)_{k = 1}^\infty$ is an enumeration of the $1s$ in the classical Thue-Morse sequence. We also establish a variant of van der Corput's Difference Theorem that is connected to unitary operators with continuous spectrum. Lastly, we obtain a new characterization of those sequence $(x_n)_{n = 1}^\infty \subseteq [0,1]$ for which $(x_{n+h},x_n)_{n = 1}^\infty$ is uniformly distributed in $[0,1]^2$ for all $h \in \mathbb{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_01123 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A generalization of van der Corput's Difference Theorem Farhangi, Sohail Dynamical Systems 37A25, 11K06, 11K38, 37A30 We prove a generalization of van der Corput's Difference Theorem in the theory of uniform distribution by establishing a connection with unitary operators that have Lebesgue spectrum. This allows us to show, for example, that if $(x_n)_{n = 1}^\infty \subseteq [0,1]$ is such that $(x_{n+h}-x_n)_{n = 1}^\infty$ is uniformly distributed for all $h \in \mathbb{N}$, then $(x_{n_k})_{k = 1}^\infty$ is uniformly distributed, where $(n_k)_{k = 1}^\infty$ is an enumeration of the $1s$ in the classical Thue-Morse sequence. We also establish a variant of van der Corput's Difference Theorem that is connected to unitary operators with continuous spectrum. Lastly, we obtain a new characterization of those sequence $(x_n)_{n = 1}^\infty \subseteq [0,1]$ for which $(x_{n+h},x_n)_{n = 1}^\infty$ is uniformly distributed in $[0,1]^2$ for all $h \in \mathbb{N}$. |
| title | A generalization of van der Corput's Difference Theorem |
| topic | Dynamical Systems 37A25, 11K06, 11K38, 37A30 |
| url | https://arxiv.org/abs/2106.01123 |