Timeline for The supremum value of $\int f(t) \log{\frac{1}{|t|}} \, dt$ for normalized Fourier pairs non-negative outside of $[-1,1]$
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15 events
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| Nov 16, 2017 at 0:31 | comment | added | Vesselin Dimitrov | One minor comment. I think that, by the triangle inequality, the positivity $a,b \geq 0$ is irrelevant in the forbidding relation $\alpha \delta + \beta = a + \check{b}$. In the given construction, I couldn't see right away how to arrange that $a$ and $b$ are everywhere non-negative. But the relation already forces $\int_{|t| \geq 1} (|f| + |\hat{f}|) \, dt \geq \epsilon_0 > 0$ to be bounded away from $0$ under the condition $f(0) = \hat{f}(0) = 1$. | |
| Nov 13, 2017 at 20:17 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| Nov 13, 2017 at 16:33 | comment | added | Vesselin Dimitrov | I see. I was expecting this, since otherwise sharpness would contain 'for free' a proof of the PNT and raise the possibility of quantiative refinements, which seems just too good. It is the uncertainty principle that prevents us from reaching PNT this way. I will now study this technique and paper in detail, but it makes sense. This is extremely helpful, thank you very much! | |
| Nov 13, 2017 at 15:51 | comment | added | Terry Tao | You're right - the two seminorms we have here are strong enough to combine to a norm that allows the uncertainty principle to kick in, so in fact my answer should be reversed - some improvement to $1+\gamma$ is in fact possible. I've updated the answer accordingly. | |
| Nov 13, 2017 at 15:50 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| Nov 13, 2017 at 7:25 | comment | added | Vesselin Dimitrov | As $f, \widehat{f}$ are assumed non-negative on $\mathbb{R} \setminus [-1,1]$, aren't we implicitly dealing with their $L^1$ norms on $\mathbb{R} \setminus [-1,1]$? | |
| Nov 13, 2017 at 7:02 | comment | added | Vesselin Dimitrov | There is also a complementary lower bound. If $f$ has the same normalization condition ($f(0) = \widehat{f}(0) = 1$), but instead $f, \widehat{f} \leq 0$ outside of $[-1,1]$, then the same argument gives $\int f(t) \log{\frac{1}{|t|}} \, dt > 1 + \gamma$. Anyway, a sharp upper bound would alone suffice for PNT, assuming of course it holds for all $X$ without exceptions. | |
| Nov 13, 2017 at 6:23 | comment | added | Terry Tao | Hang on, there is an issue with the last part of my argument - $\check b$ and the antiderivative of $\hat a$ are not in fact continuous at the origin, so I can't yet rule out a non-trivial solution to the equation $\alpha \delta + \beta = a + \check b$. I'll have to think about this. | |
| Nov 13, 2017 at 6:05 | comment | added | Terry Tao | The uncertainty principle will kick in if one has some control on $f$ in some function space norm (e.g. $L^2$ norm, $L^p$ norm, Sobolev norm, etc.). It's only because one has no norm control here that one can evade the uncertainty principle. It's odd though that this argument seems to give "one half of the prime number theorem" in some sense. When combined with Brun-Titchmarsh to estimate the error terms, it seems to give an elementary proof of PNT that is slightly different from the Erdos-Selberg one. | |
| Nov 13, 2017 at 4:41 | comment | added | Vesselin Dimitrov | This is amazing to me, since I rather expected that some form of the uncertainty principle would have to give a negative answer. Instead, as you explained, a positive existential answer does follow by a simple application of Hahn-Banach! I assume this means it could makes sense to try to search for a new approach to the prime number theorem by attempting to construct an explicit sequence of $f$ (or, if this is easier, the $g$ of my post), that realize the $1+\gamma$ supremum value? Any such sequence would place an explicit and sharp estimate of the prime number sum $S(X)$. | |
| Nov 13, 2017 at 4:27 | vote | accept | Vesselin Dimitrov | ||
| Nov 13, 2017 at 3:32 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| Nov 13, 2017 at 3:22 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| Nov 13, 2017 at 3:11 | history | edited | Terry Tao | CC BY-SA 3.0 |
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| Nov 13, 2017 at 3:00 | history | answered | Terry Tao | CC BY-SA 3.0 |