Timeline for Does Weyl's Inequality prove equidistribution?
Current License: CC BY-SA 4.0
23 events
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| May 18 at 8:52 | history | edited | Martin Sleziak |
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| Mar 26 at 15:38 | history | edited | GH from MO |
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| S Mar 26 at 15:23 | history | suggested | The Amplitwist | CC BY-SA 4.0 |
fixed broken link to Wikipedia; added link to George Lowther's comment
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Aug 23, 2010 at 11:26 | history | edited | David E Speyer | CC BY-SA 2.5 |
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| Aug 23, 2010 at 11:25 | comment | added | David E Speyer | I think you are right. I didn't check the details of Gower's proof, but I'm pretty sure the exponent should be $1/2^{d-1}$, not $1/(2^d -1)$ | |
| Aug 23, 2010 at 1:24 | comment | added | George Lowther | Do Gower's notes contain a typo, reproduced in your version of Weyl's inequality? The exponent of $1/(2^d-1)$ is different from that stated in the Wikipedia entry, which is $2^{1-d}$. Also, looking at Gower's proof of Weyl's inequality in Theorem 3.6 of the notes you link, it does seem like he has proved it for an exponent of $2^{1-d}$ but has a typo in the main statement of the result. | |
| Aug 18, 2010 at 15:50 | answer | added | George Lowther | timeline score: 2 | |
| Aug 18, 2010 at 15:16 | comment | added | David E Speyer | If you put it as an answer here, it is probably a bit more likely to turn up on google searches. But either way is fine by me. | |
| Aug 18, 2010 at 14:20 | comment | added | George Lowther | I can construct examples, but they're a bit complicated to fit as a comment. Should I post it as an answer, even though it is not answering your original question? Otherwise I can post to my blog and link there. | |
| Aug 18, 2010 at 14:09 | comment | added | David E Speyer | I no longer believe I know such a bound to be true. However, that doesn't mean I see yet how to show that I can make $S_N/N$ go to zero as slowly as I like, in the sense you claim. (It is easy to see that for any $N$ and any $a<1$, there is a $\theta$ such that $S_N/N > a$. But that is not the order of quantifiers I care about.) I'd love to see the details if you have an argument for this. | |
| Aug 18, 2010 at 13:22 | comment | added | George Lowther | In fact, I don't think it is true even for d=2. By choosing $\theta$ very close to a rational, you can make $S_N/N=\frac1N\sum_{n=1}^n\exp(2\pi i\theta n^2)$ tend to zero as slowly as you like. That is, for any $h\colon\mathbb{N}\to\mathbb{R}$ tending to zero, $\theta$ can be chosen such that $\limsup S_N/(Nh(N))=\infty$. So the equidistribution theorem can't be strengthened without putting further restrictions on how well θ can be approximated by rationals. | |
| Aug 18, 2010 at 12:39 | comment | added | George Lowther | In the math.SE question you say that $S_N/N=O(N^{\epsilon-1/2^d})$. Is it even true? Do you have any reference for this? The proof below by Benoit doesn't seem to extend to give a convergence rate, and Terry Tao's proof doesn't say how fast $S_N/N$ goes to zero. | |
| Aug 18, 2010 at 11:18 | vote | accept | David E Speyer | ||
| Aug 18, 2010 at 11:18 | vote | accept | David E Speyer | ||
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| Aug 18, 2010 at 9:35 | answer | added | Benoît Kloeckner | timeline score: 6 | |
| Aug 17, 2010 at 22:33 | answer | added | George Lowther | timeline score: 3 | |
| Aug 17, 2010 at 22:25 | comment | added | David E Speyer | I actually got into this because I was trying to answer the math.SE question I linked, so I needed a strengthened version of $S_N/N \to 0$. I thought Weyl's inequality was what I was looking for, and was then very confused when I realized I couldn't even use it to get that far. | |
| Aug 17, 2010 at 22:12 | comment | added | Helge | Why prove equidistribution of this sequence using Weyl's inequality? Der Van der Corput lemma (Corollary 2 in terrytao.wordpress.com/2008/06/14/…) provides a much simpler proof. And as you observe yourself, the conclusions of Weyl's inequality here are not very good, which they can't be, since for the numbers you describe, the sequence $\theta n$ is very close to being periodic. | |
| Aug 17, 2010 at 20:55 | answer | added | Ben Green | timeline score: 7 | |
| Aug 17, 2010 at 20:49 | history | edited | David E Speyer | CC BY-SA 2.5 |
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| Aug 17, 2010 at 20:14 | history | asked | David E Speyer | CC BY-SA 2.5 |