Timeline for Function on $\mathbb{Z}/p^k \mathbb{Z}$ with small Fourier transform?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Mar 2, 2023 at 21:29 | comment | added | H A Helfgott | I was hoping for a simpler answer, but technically it does. | |
| Mar 2, 2023 at 18:47 | comment | added | kodlu | Does the Rudin Shapiro polynomial construction (which is explicit though maybe not "algebraic") answer your original question? | |
| Mar 1, 2023 at 17:06 | comment | added | H A Helfgott | Martin just told me that he learned about this question from Afonso Bandeira (ETHZ). | |
| Mar 1, 2023 at 14:30 | comment | added | H A Helfgott | Right. Here is the link: ams.org/journals/tran/1985-289-02/S0002-9947-1985-0784009-0/… | |
| Mar 1, 2023 at 14:19 | comment | added | Ben Green | I see, yes indeed it follows immediately (but not explicitly) from the Spencer result. | |
| Mar 1, 2023 at 14:06 | comment | added | H A Helfgott | @BenGreen I was replying to Joe Silverman, sorry. Thanks for the reference. On "existence is not hard" - perhaps, instead of "not hard" I should have said "immediate, given a standard result that I did not know about until a couple of days ago, when Martin Kassabov told me about it" - I mean J. Spencer, "Six standard deviations suffice", TAMS v289 n 2 (1986). You are right that the obvious probabilistic approach would miss by a $\log$ factor. Also, thanks to Martin K. for the question, which is really his! | |
| Mar 1, 2023 at 13:59 | comment | added | Ben Green | Not sure I understand your comment Harald. If one uses the results of Balister arxiv.org/abs/1909.08777 then you can take $n = p^k$ and this will give an example; the exponential sum is actually uniformly small at all $\theta$, not just $\theta = r/p^k$. I'll have to think whether there is an example that is more natural as regards the $p$-structure on $\mathbf{Z}/p^k \mathbf{Z}$. | |
| Mar 1, 2023 at 12:49 | history | edited | H A Helfgott | CC BY-SA 4.0 |
added 2 characters in body
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| Mar 1, 2023 at 12:47 | comment | added | H A Helfgott | I meant $p^k$ in the denominator - thanks. | |
| Mar 1, 2023 at 11:56 | comment | added | Ben Green | A construction based on Rudin-Shapiro polynomials should work, see the introduction to Flat Littlewood polynomials exist by Paul Balister, Bela Bollobas, Robert Morris, Julian Sahasrabudhe and Marius Tiba. By the way, why is "existence not hard"? A random construction would miss by a log factor. | |
| Mar 1, 2023 at 11:10 | comment | added | Joe Silverman | Does $x$ run over $\mathbb Z/p\mathbb Z$ or $\mathbb Z/p^k\mathbb Z$ in $\hat f(\xi)$? And do you really mean $e(-\xi x/p)$, or should it be $e(-\xi x/p^k)$, with $x$ running over $\mathbb Z/p^k\mathbb Z$? | |
| Mar 1, 2023 at 7:59 | history | asked | H A Helfgott | CC BY-SA 4.0 |