Timeline for Fourier decay rate of Cantor measures
Current License: CC BY-SA 3.0
15 events
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| Feb 11, 2020 at 21:10 | comment | added | Pablo Shmerkin | @JasonJones . Just read your comment. I don't have a reference at hand for (1), will let you know if I find one. (2) is not known. Although it is known that power Fourier decay is the typical behaviour, there are essentially no explicit parameters for which power Fourier decay is known to hold. | |
| Jan 12, 2020 at 9:01 | comment | added | JasonJones | I have some questions about your remark on Salem numbers. (1) Do have a reference for this? I searched through Salem's "Algebraic Numbers and Fourier Analysis" and I also looked for likely candidates among all the papers citing that book. But I had no luck. (2) Is it true that $\theta^{-1}$ is a Salem number if and only if $C_{\theta}$ supports no measure with power Fourier decay? I know $\theta^{-1}$ is a PV number if and only if $C_{\theta}$ supports no measure whose Fourier transform decays to $0$. | |
| Jun 2, 2012 at 18:53 | comment | added | Pablo Shmerkin | The result is only for self-similar sets, i.e. the relative positions are the same at all steps (they are only free at the first step). So it doesn't contradict Bluhm's result... adding randomness kills the resonances so changes everything. See Theorem 4 of "Resonance between Cantor sets" for the statement for self-similar sets which implies they cannot be Salem (when the logs of the ratios are commensurable). | |
| Jun 2, 2012 at 16:37 | comment | added | Syang Chen | That would be nice. Could you tell me precisely in which paper can I find the result? If I understand correctly, it contradicts Bluhm's result (Theorem 5). In his construction, the ratios are of the form $\theta^k$, only the positions are randomized. | |
| Jun 2, 2012 at 11:44 | comment | added | Pablo Shmerkin | The argument extends to self-similar sets in which the logs of the contraction ratios forms an arithmetic set, in particular when they are all equal, regardless of positions (see my paper with Y. Peres for details). If there are two rationally incommensurable ratios, I believe nothing is known about Fourier decay. About Bluhm's paper, I haven't read the proof in detail but the strategy seems sound, and it's a natural result. | |
| Jun 2, 2012 at 3:55 | comment | added | Syang Chen | Nice. So generally the "neat" Cantor sets (with neat positions, ratios) can not be Salem? | |
| Jun 2, 2012 at 3:39 | comment | added | Syang Chen | The paper proves that the randomized Cantor set (keeping the dissection number constant 2) is almost surely Salem. But I couldn't quite go through the proof. Maybe I was just confused... | |
| Jun 2, 2012 at 3:29 | vote | accept | Syang Chen | ||
| Jun 2, 2012 at 3:40 | |||||
| Jun 1, 2012 at 8:36 | history | edited | Pablo Shmerkin | CC BY-SA 3.0 |
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| Jun 1, 2012 at 8:33 | comment | added | Pablo Shmerkin | I haven't read it in detail but can have a look, what is the question? | |
| Jun 1, 2012 at 8:24 | history | edited | Pablo Shmerkin | CC BY-SA 3.0 |
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| Jun 1, 2012 at 8:12 | history | edited | Pablo Shmerkin | CC BY-SA 3.0 |
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| Jun 1, 2012 at 6:37 | comment | added | Syang Chen | Are you familiar with the paper you mentioned above? I have a question about (3.3) in that paper. | |
| May 31, 2012 at 12:08 | history | edited | Pablo Shmerkin | CC BY-SA 3.0 |
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| May 31, 2012 at 12:02 | history | answered | Pablo Shmerkin | CC BY-SA 3.0 |