Prove the following limits of a sequence of sets?
(See the paper in the attatchment for a background behind the problems.)
Suppose $A=\mathbb{Q}$ where $f:A\to\mathbb{R}$ such that:
$f(x)= \begin{cases} 1 & x\in\left\{{(2n+1)}/{2m}:n\in\mathbb{Z}, m\in\mathbb{N}\right\}\\ 0 & x\not\in\left\{{(2n+1)}/{2m}:n\in\mathbb{Z}, m\in\mathbb{N}\right\}\\ \end{cases} $
and, in sequel, the integral of $f$ is taken w.r.t to the counting measure.
1. If $(F_r)_{r\in\mathbb{N}}= \left(\left\{c/r!:c\in\mathbb{Z},\, -r\cdot r!\le c\le r\cdot r! \right\}\right)_{r\in\mathbb{N}}$ prove that:
$$\forall(\epsilon>0)\exists(N\in\mathbb{N})\forall(r\in\mathbb{N})\left(r\ge N\Rightarrow\left|\frac{1}{\left|F_r\right|}\int_{F_r}f(x)\, d\mathbf{x}-1\right|< \epsilon\right)$$
2. If $(F_r)_{r\in\mathbb{N}}= \left(\left\{c/d:c\in\mathbb{Z},\, d\in\mathbb{N},\, d\le r,\, -dr\le c\le dr \right\}\right)_{r\in\mathbb{N}}$ prove that:
$$\forall(\epsilon>0)\exists(N\in\mathbb{N})\forall(r\in\mathbb{N})\left(r\ge N\Rightarrow\left|\frac{1}{\left|F_r\right|}\int_{F_r}f(x)\, d\mathbf{x}-1/3\right|< \epsilon\right)$$
Bharathk98
Answer
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2 Attachments
M F H
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Instead {F_r}r can I write (F_r)r?
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Yes I think that would be OK (of course, also "(...)" on the R.H.S.)
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I know you cannot share latex documents online but perhaps in future answers you can post latex pdfs (e.g. register in latex overleaf for free). Registering is actually pretty simple and should make organizing mathematical ideas a million times easier.
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I have often tried to type my answer with math formulae in the provided window, but then I have lost much much time due to lost formatting and special characters (math symbols ,greek letters , ....) so I have given up that. I can try to copy -paste but I don't want to retype all the math stuff in LaTeX. Oh, I also have an overleaf account , but well, I find google docs even more convenient.... :-)
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I’m not sure how I could all the steps into my research paper. Perhaps I need a simpler example or you can give a concise version of the document which I could transfer to my research paper. I will give the longer version to my tutor and see what he thinks.
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OK, I tried to be verbose because previously I have often be asked to elaborate on many details. Anyway, you can cut out the first part where I clarify notations, and of course the 2 Appendices aren't needed either. The result should be very concise. If you wish I can make a shorter "summary" version.
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@MFH Sure, you can make a summary version.
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OK, I made a short version of 10+15 lines (I don't think one can make it shorter) in the online version, see https://docs.google.com/document/u/3/d/e/2PACX-1vRYDHIFG9J7zqMP6hnxqEwoLOPichTOTa_SwFPmCayLwfE0u74A1-LwOfrNg6L285-Ri4DQT64ycEcX/pub If you wish I can add a PDF of that.
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IDK whether this is OK but I could also give you access to the google doc which you might be able to copy to your account...?
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I will try to write the proof in my own words. You don’t have to give me access to the google account.
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I will also give you credit if I have to include a proof of my assumptions.
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OK ! Feel free to ask if anything is unclear.
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Low bounty!
How much should it be?
Well, this looks like a tricky question to me. I think it should be at least doubled or tippled. Think about how long it would take someone to answer, it would give you an idea.
I will double it. I think for a highly skilled mathematician this should be an exercise.
Yes, but highly skilled mathematicians also tend to value their time. I'm busy at the moment, but I may have some time over the weekend to think about it, if no one else answers by then.
Very low bounty! Highly skilled Mathematicians charge at least $100/hour and this question needs at least 1-2 hours work. So, figure out a fair bounty.
@Kav10 PaulF says he will look at it over the weekend. If it’s too complex, I’ll charge more.
How do you know those are the correct values for the limits?
@Mathe I used computer programming. I think I’m correct.
The statement is ill formulated. It starts "Let f = ... where the integral of f is taken w.r.t to the counting measure.", But there ("where") is no integral.
Also, the definition "If { F_r}_r = { set(r) }_r ...." is not a definition. Example: { F_1, F_2, F_3 ...} = { {A}, {B}, {C}, ...} : By the definition of a set, the right hand side is strictly equal to { {B}, {A}, {C}, ...} and any other permutation of what is written there. There is no order in a set, so that equation does not define F_r.
Also, according to the current definition, the sets F_r contain only one number, namely, one must have c = - r * r! (left and right member of the double inequality is the same !) and so F_r = { -r }. But I can guess what is meant and try to answer will a minimum of necessary corrections of what is actually asked.
@MFH I made the necessary corrections to the first sequence of sets. How should I (instead) defined {F_r}r in general.
I think a simple and clear way would be : ∀ r ∈ ℕ : F_r = { c / r ! : … } . I propose this and other clarification my answer (under way, but I'm only at the beginning...)
(The first part, with |c| < r.r!, is actually easy and I've written up the complete proof, the second part is trickier. If you are interested I can provide you a link to an online document with the solution, but I think it should be done in a "private" comment.)
@Matt I made changes to the question per you clarifications. Let me know if it’s clear enough.
@MFH Sure. I’m fine with a private comment.