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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.12763 |
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| _version_ | 1866916579106619392 |
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| author | Fruehwirth, Lorenz Hauke, Manuel |
| author_facet | Fruehwirth, Lorenz Hauke, Manuel |
| contents | Given a lacunary sequence $(n_k)_{k \in \mathbb{N}}$, arbitrary positive weights $(c_k)_{k \in \mathbb{N}}$ that satisfy a Lindeberg-Feller condition, and a function $f: \mathbb{T} \to \mathbb{R}$ whose Fourier coefficients $\hat{f_k}$ decay at rate $\frac{1}{k^{1/2 + \varepsilon}}$, we prove central limit theorems for $\sum_{k \leq N}c_kf(n_kx)$, provided $(n_k)_{k \in \mathbb{N}}$ satisfies a Diophantine condition that is necessary in general. This addresses a question raised by M. Kac [Ann. of Math., 1946]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12763 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Problem of Kac concerning Anisotropic Lacunary Sums Fruehwirth, Lorenz Hauke, Manuel Probability Dynamical Systems Number Theory Primary 42A55, 60F05, Secondary 11D04, 11D45 Given a lacunary sequence $(n_k)_{k \in \mathbb{N}}$, arbitrary positive weights $(c_k)_{k \in \mathbb{N}}$ that satisfy a Lindeberg-Feller condition, and a function $f: \mathbb{T} \to \mathbb{R}$ whose Fourier coefficients $\hat{f_k}$ decay at rate $\frac{1}{k^{1/2 + \varepsilon}}$, we prove central limit theorems for $\sum_{k \leq N}c_kf(n_kx)$, provided $(n_k)_{k \in \mathbb{N}}$ satisfies a Diophantine condition that is necessary in general. This addresses a question raised by M. Kac [Ann. of Math., 1946]. |
| title | On a Problem of Kac concerning Anisotropic Lacunary Sums |
| topic | Probability Dynamical Systems Number Theory Primary 42A55, 60F05, Secondary 11D04, 11D45 |
| url | https://arxiv.org/abs/2501.12763 |