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Sep
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reviewed | Approve locally constant constructible sheaves and finite etale coverings |
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Sep
15 |
comment |
Smoothed exponential sums: bounds and sources?
(obviously, $|\widehat{f^{(4)}}|_1$ above should be $|\widehat{f^{(4)}}|_\infty$.) |
Sep
14 |
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Smoothed exponential sums: bounds and sources?
In summary (given the answer below and other comments): we can do better than the above for $k\geq 4$ (and better than Tao for $k\geq 2$, and the proposed inequality for $k=1$ isn't true. But what about Tao's bound for $k=1$? Was this known before? (Note that it is precisely what one gets by applying bounds for $\sum e(\alpha n)$ (non-smoothed) over intervals - not that this is genuinely different from Tao's proof by summation by parts.) |
Sep
14 |
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Smoothed exponential sums: bounds and sources?
- and, of course, we cannot drop the absolute values for $k$ odd (as far as I can see). So all I know how to do then is to deduce a bound for $k=2r+1$ from the bound for $k=2r$, as I outlined in my reply to my own question. |
Sep
14 |
awarded | Nice Question |
Sep
14 |
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Smoothed exponential sums: bounds and sources?
Ha-ha. Just to complete what I was saying: taking derivatives, we obtain $\sum_{n=-\infty}^{n=\infty} \frac{1}{(n+s)^4} =\frac{\pi^4}{(\sin s\pi)^4} (1 -\frac{2}{3} \sin^2 s \pi)$, and so on; hence, for $k=4$, $\frac{|\widehat{f^{(4)}}|_1}{(2 \sin \pi \alpha)^4}$ can be replaced by $\frac{1 - 2 (\sin \pi \alpha)^2/3}{(2 \sin \pi \alpha)^4} |\widehat{f^{(4)}}|_1$, and so on. |
Sep
14 |
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Smoothed exponential sums: bounds and sources?
On $k\geq 2$: to obtain the bound I gave for $k=2$, I start as you do (I think we are normalizing the Fourier transform differently) and then I use the identity $\sum_{n=-\infty}^\infty \frac{1}{(n+s)^2} = \frac{\pi^2}{(\sin s\pi)^2}$ (a consequence of a formula of Euler's). |
Sep
14 |
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Smoothed exponential sums: bounds and sources?
On $k=1$: nice! |
Sep
14 |
revised |
Smoothed exponential sums: bounds and sources?
added 31 characters in body |
Sep
14 |
revised |
Smoothed exponential sums: bounds and sources?
edited title |
Sep
14 |
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Smoothed exponential sums: bounds and sources?
A partial answer to my own question: the bound I give for $k=2$ implies similar bounds for $k>2$. This goes as follows: by summation by parts, $$\sum_n (f(n+1)-f(n)) e(\alpha n) = \sum_n f(n) (e(\alpha n)- e(\alpha (n+1))) = (1-e(-\alpha)) \sum_n f(n) e(\alpha n);$$ hence, $\sum_n f(n) e(\alpha n) = \frac{1}{1-e(-\alpha)} \int_0^1 \left(\sum_n f'(n+t) e(\alpha n)\right) dt$. Now use the inequality for $k=2$, with $f'(x+t)$ instead of $f(x)$; we obtain that $|\sum_n f(n) e(\alpha n)|\leq |\widehat{f^{(3)}}|_\infty/|2 \sin(\pi \alpha)|^3$. Iterate to obtain the result for all $k>2$. |
Sep
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asked | Smoothed exponential sums: bounds and sources? |
Jun
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awarded | Popular Question |
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20 |
awarded | Nice Question |
Apr
16 |
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A variant of Goldbach Conjecture
It can, but it could be a little tricky - most of the weights I use are not compactly supported (though they decay very fast -- they are almost supported on a compact interval). |
Apr
15 |
awarded | Good Answer |
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15 |
awarded | Mortarboard |
Apr
15 |
awarded | Nice Answer |