Timeline for Overconcentration of Poles on the Circle of Convergence of a Power Series with Bounded Coefficients
Current License: CC BY-SA 4.0
30 events
| when toggle format | what | by | license | comment | |
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| Jun 21, 2019 at 20:22 | comment | added | MCS | Here you go: mathoverflow.net/questions/334532/… | |
| Jun 21, 2019 at 19:55 | comment | added | fedja | @MCS OK, then think of what (im)possibilities interest you most and start a new thread. This one is cluttered enough IMHO ;-) | |
| Jun 21, 2019 at 1:58 | comment | added | MCS | I'm using these functions to investigate matters in number theory. I call these functions "set-series", and use their analytical properties to deduce information about the set $V$ that they model. Knowing what is possible for generic set-series informs the kinds of arguments I can('t) make. For example, my original question about the discreteness of $T_{V}$ has to do with the fact that I have an argument which works for set-series with that property. Dealing with generic cases, it helps to know what the possibilities are, rather than just taking things for granted. :) | |
| Jun 21, 2019 at 1:01 | comment | added | fedja | @MCS Have you read "Alice in Wonderland"? (or was it "Through the looking glass"?). There is a passage there where she recites "Father William" and I like the ending a lot ;-)) That was a joke, of course, but seriously, step back for a while and spend some time understanding what you really want to get and where it is all going. Once you have a clear picture of what you are after, I'll be happy to think of some concise question you may have (start a new thread then) but now I'd rather fry some other fish. Still, you can always ask for clarifications on what has been posted already :-). | |
| Jun 21, 2019 at 0:39 | comment | added | MCS | This is awesome; you are awesome. I'll have to spend time going through this example in detail. That being said, I hope you don't mind if I take you up on your offer to ask more questions. :) For an arbitrary $V$, what can be said about the set of $t$ (in $\mathbb{Q}$, and in $\mathbb{R}$) for which $R\left(t\right)$ does or does not exist? For example, does $\mathbb{Z}\subseteq T_{V}$ imply $\mathbb{Q}\subseteq T_{V}$?—or, can there be infinitely many points (in $\mathbb{Q}$? in $\mathbb{R}$?) where $R\left(t\right)$ does not exist? | |
| Jun 18, 2019 at 1:55 | comment | added | fedja | @MCS Posted.... | |
| Jun 18, 2019 at 1:50 | history | edited | fedja | CC BY-SA 4.0 |
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| Jun 17, 2019 at 23:22 | comment | added | MCS | Take your time. :) | |
| Jun 17, 2019 at 23:18 | comment | added | fedja | @MCS The last condition is not a restriction at all. OK, I will post the example today but somewhat later (have other fish to fry at the moment) | |
| Jun 17, 2019 at 22:29 | comment | added | MCS | Post it and let me take a look-see. :D Edit: Actually, wait. There is one more condition I can impose: V contains an infinite arithmetic progression (b, a+b, 2a+b, ...). | |
| Jun 17, 2019 at 21:20 | comment | added | fedja | @MCS At this moment I can construct an example when 1) $R(t)$ exists for all $t$, 2) $R(2t)=R(t+\frac 12)+R(t)$ for all $t$, 3) There is a sequence $t_j$ of rational numbers tending to $0$ such that $R(t_j)\ne 0$. If this closes the story, I'll post it. Otherwise let me know if there is anything else that you know about $V$. :-) | |
| Jun 17, 2019 at 19:26 | comment | added | MCS | Just to make sure you haven't misunderstand me: my argument makes no assumptions about whether or not there are irrational values of $t$ for which $R\left(t\right)$ exists. Like in your original counterexample, there could very well be a non-discrete set of irrational values of $t$ for which $R\left(t\right)$ exists and is non-zero, to boot. All I want is the discreteness of the set of rational values of $t$ for which $R\left(t\right)$ exists and is non-zero. My hope was that the fact that $\psi_{V}\left(z\right)'s$ coefficients were only 0 or 1 would be enough to prove this. | |
| Jun 17, 2019 at 6:36 | comment | added | fedja | @MSC Unfortunately, just assuming rationality is not enough. Your property should be crucial in some way if the desired result holds. It will take me some time to comprehend the meaning of it... | |
| Jun 17, 2019 at 6:03 | comment | added | MCS | Also, we can assume that $R\left(0\right)$ exists and is non-zero. And, just to be clear, $T_{V}$ is still a subset of $\mathbb{Q}$. (Okay, self, time for bed now. xD) | |
| Jun 17, 2019 at 5:53 | comment | added | MCS | It might be worth mentioning that this is just the simplest case; the general case is: $$R\left(t\right)=R\left(Nt\right)-\sum_{n=1}^{N-1}e^{\frac{2n\pi i}{N}}R\left(t+\frac{n}{N}\right)$$ I'll admit: I'm curious to see what the difference would be if we went in with the asumption that each individual term exists for all $t\in T_{V}$, as opposed to what happens when it is merely assumed that both sides exist, rather than the individual terms. Although, a proof for the weaker assumptions would be best. Anyhow, thanks again. :) | |
| Jun 17, 2019 at 5:52 | comment | added | MCS | As for the non-vanshing, I goofed! $T_{V}$ is the set of all $t$ for which $R\left(t\right)$ exists; I want to show that the subset $S_{V}\subseteq T_{V}$ on which $R\left(t\right)\neq0$ is discrete (Sorry for the confusion). | |
| Jun 17, 2019 at 5:52 | comment | added | MCS | In writing, $R\left(2t\right)-R\left(t+\frac{1}{2}\right)=R\left(t\right)$ for all $t\in T_{V}$, it means that: $$\lim_{y\downarrow0}\left(iy\psi_{V}\left(iy+2t\right)-iy\psi_{V}\left(iy+t+\frac{1}{2}\right)\right)=\lim_{y\downarrow0}iy\psi_{V}\left(iy+t\right)$$ | |
| Jun 17, 2019 at 5:32 | comment | added | fedja | Erm... When you state the property the way you did, do you mean that the existence of $R(t)$ implies the existence of $R(t+\frac 12)$ and $R(2t)$ or something weaker or stronger than that? Also, we still assume that $R(t)\ne 0$ for $t\in T_V$, right? | |
| Jun 17, 2019 at 2:25 | comment | added | MCS | Change perspective from the disk to the half plane; write $\psi_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}e^{2v\pi iz}$. Instead of radial limits, consider the function $R\left(t\right)\overset{\textrm{def}}{=}\lim_{y\downarrow0}iy\psi_{V}\left(iy+t\right)$. Let $T_{V}$ denote the set of all $t\in\mathbb{Q}$ for which $R\left(t\right)$ exists, and suppose that $R\left(2t\right)=R\left(t\right)+R\left(t+\frac{1}{2}\right)$ holds for all $t\in T_{V}$. I want to show that $T_{V}$ is a discrete subset of $\mathbb{Q}$ (i.e., $T_{V}\cap\left[0,1\right]$ is finite). | |
| Jun 16, 2019 at 18:01 | comment | added | fedja | @MCS I can think of it more but nothing is guaranteed. So what exactly is your setup? | |
| Jun 16, 2019 at 16:40 | comment | added | MCS | If you're still interested in the case where $T_{V}$ contains only rational numbers... it turns out my proof didn't work. You've been tremendous helpful so far, so I don't want to impose any more than I already have, but... I'll admit, I'm pretty desperate here. No matter what, though, thanks for all your help so far. | |
| Jun 13, 2019 at 18:19 | comment | added | MCS | Brilliant! I always forget about the series analogue of Cauchy-Schwarz. All the abstract analysis that gets stuffed down our faces nowadays in school really obscures the connections between the theorems and their divers applications in "real life" situations like this. Thank you so much! | |
| Jun 13, 2019 at 17:38 | comment | added | fedja | @MCS Done. Feel free to ask questions if something is unclear. | |
| Jun 13, 2019 at 17:37 | history | edited | fedja | CC BY-SA 4.0 |
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| Jun 12, 2019 at 23:07 | comment | added | MCS | Using the other things I'm working with, I think I've gotten a proof of what I'm looking for. Let $R\left(t\right)\overset{\textrm{def}}{=}\lim_{x\uparrow1}\left(1-x\right)\varsigma_{V}\left(e^{2\pi it}x\right)$. I've been able to show that there is an infinite subset $S_{V}\subseteq T_{V}$ so that $R$ is constant on $S_{V}$. By your remark about the sum of the squares, this forces $\sum_{t\in S_{V}}\left|R\left(t\right)\right|^{2}<\infty$, and so $R\left(t\right)=0$ on $S_{V}$, a contradiction. I'd be really grateful if you could explain to me how to prove the sum of the squares is 0. :) | |
| Jun 12, 2019 at 18:49 | comment | added | MCS | [The sum of squares of limits is finite] - - - Sorry for being dense (again), but, I don't quite see how that follows. Almost periodic functions. Got it. That will help. Thank you—both for what you've done, and anything else you might still do. :) | |
| Jun 12, 2019 at 18:43 | comment | added | fedja | @MCS The sum of squares of limits is finite. For "only rational numbers" I'll have to think for a (possibly long) while. I doubt there is something exactly dealing with your question, but if you look at the literature devoted to almost periodic functions, you'll find all necessary tricks there. | |
| Jun 12, 2019 at 18:39 | vote | accept | MCS | ||
| Jun 12, 2019 at 18:39 | comment | added | MCS | How would one go about showing that $T_{V}$ must be countable? Also, what would happen if $T_{V}$ is required to contain only rational numbers? Also, also, any links or references you could give me on the matter would be really helpful. Thanks! | |
| Jun 12, 2019 at 3:24 | history | answered | fedja | CC BY-SA 4.0 |