Polynomial Fourier decay for fractal measures and their pushforwards
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913821299310592 |
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| author | Baker, Simon Banaji, Amlan |
| author_facet | Baker, Simon Banaji, Amlan |
| contents | We prove that the pushforwards of a very general class of fractal measures $μ$ on $\mathbb{R}^d$ under a large family of non-linear maps $F \colon \mathbb{R}^d \to \mathbb{R}$ exhibit polynomial Fourier decay: there exist $C,η>0$ such that $|\widehat{Fμ}(ξ)|\leq C|ξ|^{-η}$ for all $ξ\neq 0$. Using this, we prove that if $Φ= \{ φ_a \colon [0,1] \to [0,1] \}_{a \in \mathcal{A}}$ is an iterated function system consisting of analytic contractions, and there exists $a \in \mathcal{A}$ such that $φ_a$ is not an affine map, then every non-atomic self-conformal measure for $Φ$ has polynomial Fourier decay; this result was obtained simultaneously by Algom, Rodriguez Hertz, and Wang. We prove applications related to the Fourier uniqueness problem, Fractal Uncertainty Principles, Fourier restriction estimates, and quantitative equidistribution properties of numbers in fractal sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial Fourier decay for fractal measures and their pushforwards Baker, Simon Banaji, Amlan Dynamical Systems Classical Analysis and ODEs 42A38 (Primary) 28A80, 60F10 (Secondary) We prove that the pushforwards of a very general class of fractal measures $μ$ on $\mathbb{R}^d$ under a large family of non-linear maps $F \colon \mathbb{R}^d \to \mathbb{R}$ exhibit polynomial Fourier decay: there exist $C,η>0$ such that $|\widehat{Fμ}(ξ)|\leq C|ξ|^{-η}$ for all $ξ\neq 0$. Using this, we prove that if $Φ= \{ φ_a \colon [0,1] \to [0,1] \}_{a \in \mathcal{A}}$ is an iterated function system consisting of analytic contractions, and there exists $a \in \mathcal{A}$ such that $φ_a$ is not an affine map, then every non-atomic self-conformal measure for $Φ$ has polynomial Fourier decay; this result was obtained simultaneously by Algom, Rodriguez Hertz, and Wang. We prove applications related to the Fourier uniqueness problem, Fractal Uncertainty Principles, Fourier restriction estimates, and quantitative equidistribution properties of numbers in fractal sets. |
| title | Polynomial Fourier decay for fractal measures and their pushforwards |
| topic | Dynamical Systems Classical Analysis and ODEs 42A38 (Primary) 28A80, 60F10 (Secondary) |
| url | https://arxiv.org/abs/2401.01241 |