Self-similar and self-conformal measures with slow Fourier decay
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915777797423104 |
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| author | Baker, Simon Banaji, Amlan |
| author_facet | Baker, Simon Banaji, Amlan |
| contents | Given any function $ϕ\colon [0,\infty)\to (0,1]$ satisfying $\lim_{ξ\to\infty}ϕ(ξ) = 0$, we prove the existence of i) self-similar measures and ii) nonlinear $C^{\infty}$ self-conformal measures which are Rajchman and whose Fourier transform $\widehatμ$ satisfies \[ \limsup_{ξ\to\infty}\frac{|\widehatμ(ξ)|}{ϕ(ξ)}>0.\] Moreover, we derive new sufficient conditions for a self-conformal measure to be Rajchman, and construct an explicit self-similar measure $μ$ such that $μ$ almost every $x$ is normal in base $10$ but the sequence $(10^{n}x \mod 1)_{n=1}^{\infty}$ equidistributes extremely slowly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05593 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Self-similar and self-conformal measures with slow Fourier decay Baker, Simon Banaji, Amlan Dynamical Systems Classical Analysis and ODEs Number Theory 42A38 (Primary) 28A80, 11J71 (Secondary) Given any function $ϕ\colon [0,\infty)\to (0,1]$ satisfying $\lim_{ξ\to\infty}ϕ(ξ) = 0$, we prove the existence of i) self-similar measures and ii) nonlinear $C^{\infty}$ self-conformal measures which are Rajchman and whose Fourier transform $\widehatμ$ satisfies \[ \limsup_{ξ\to\infty}\frac{|\widehatμ(ξ)|}{ϕ(ξ)}>0.\] Moreover, we derive new sufficient conditions for a self-conformal measure to be Rajchman, and construct an explicit self-similar measure $μ$ such that $μ$ almost every $x$ is normal in base $10$ but the sequence $(10^{n}x \mod 1)_{n=1}^{\infty}$ equidistributes extremely slowly. |
| title | Self-similar and self-conformal measures with slow Fourier decay |
| topic | Dynamical Systems Classical Analysis and ODEs Number Theory 42A38 (Primary) 28A80, 11J71 (Secondary) |
| url | https://arxiv.org/abs/2602.05593 |