Intersection of integers and open cover of some arithmetic progression
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Question: Given an arithmetic progression $$(a_n)_{n=1}^{infty}={a,apm d,apm 2d,dots}$$ where $a,dinBbb R$ and given $delta>0$. What can one say about the cardinality of the set$$Bbb Zcapbigcup_{n=1}^{infty}B(a_n,delta)$$? Is it empty, finite, or infinite? Possibly the case will be more interesting if $a,dinBbb Q^c$. In other words, given any AP, can we subtract a subsequence of this AP such that the decimal part is close to $0$ or $1$?
($B(a_n,delta)$ is the open ball centered at $a_n$ with radius $delta$)
real-analysis
asked Dec 4 at 16:22
Nick
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Question: Given an arithmetic progression $$(a_n)_{n=1}^{infty}={a,apm d,apm 2d,dots}$$ where $a,dinBbb R$ and given $delta>0$. What can one say about the cardinality of the set$$Bbb Zcapbigcup_{n=1}^{infty}B(a_n,delta)$$? Is it empty, finite, or infinite? Possibly the case will be more interesting if $a,dinBbb Q^c$. In other words, given any AP, can we subtract a subsequence of this AP such that the decimal part is close to $0$ or $1$?
($B(a_n,delta)$ is the open ball centered at $a_n$ with radius $delta$)
real-analysis
asked Dec 4 at 16:22
Nick
1,6401417
I can only say, that it can be finite and non-empty. In other words, it is either countably infinite or empty. (Both cases are possible, but it won't depend solely on choice of $delta$).
– kolobokish
Dec 4 at 17:02
Sorry. I misspelled. It "can't" be finite and non-empty.
– kolobokish
Dec 4 at 17:56
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Question: Given an arithmetic progression $$(a_n)_{n=1}^{infty}={a,apm d,apm 2d,dots}$$ where $a,dinBbb R$ and given $delta>0$. What can one say about the cardinality of the set$$Bbb Zcapbigcup_{n=1}^{infty}B(a_n,delta)$$? Is it empty, finite, or infinite? Possibly the case will be more interesting if $a,dinBbb Q^c$. In other words, given any AP, can we subtract a subsequence of this AP such that the decimal part is close to $0$ or $1$?
($B(a_n,delta)$ is the open ball centered at $a_n$ with radius $delta$)
real-analysis
asked Dec 4 at 16:22
Nick
1,6401417
Question: Given an arithmetic progression $$(a_n)_{n=1}^{infty}={a,apm d,apm 2d,dots}$$ where $a,dinBbb R$ and given $delta>0$. What can one say about the cardinality of the set$$Bbb Zcapbigcup_{n=1}^{infty}B(a_n,delta)$$? Is it empty, finite, or infinite? Possibly the case will be more interesting if $a,dinBbb Q^c$. In other words, given any AP, can we subtract a subsequence of this AP such that the decimal part is close to $0$ or $1$?
($B(a_n,delta)$ is the open ball centered at $a_n$ with radius $delta$)
real-analysis
real-analysis
asked Dec 4 at 16:22
Nick
1,6401417
asked Dec 4 at 16:22
Nick
1,6401417
edited Dec 4 at 16:30
asked Dec 4 at 16:22
Nick
1,6401417
asked Dec 4 at 16:22
Nick
1,6401417
asked Dec 4 at 16:22
Nick
1,6401417
1,6401417
I can only say, that it can be finite and non-empty. In other words, it is either countably infinite or empty. (Both cases are possible, but it won't depend solely on choice of $delta$).
– kolobokish
Dec 4 at 17:02
Sorry. I misspelled. It "can't" be finite and non-empty.
– kolobokish
Dec 4 at 17:56
add a comment |
I can only say, that it can be finite and non-empty. In other words, it is either countably infinite or empty. (Both cases are possible, but it won't depend solely on choice of $delta$).
– kolobokish
Dec 4 at 17:02
Sorry. I misspelled. It "can't" be finite and non-empty.
– kolobokish
Dec 4 at 17:56
I can only say, that it can be finite and non-empty. In other words, it is either countably infinite or empty. (Both cases are possible, but it won't depend solely on choice of $delta$).
– kolobokish
Dec 4 at 17:02
I can only say, that it can be finite and non-empty. In other words, it is either countably infinite or empty. (Both cases are possible, but it won't depend solely on choice of $delta$).
– kolobokish
Dec 4 at 17:02
Sorry. I misspelled. It "can't" be finite and non-empty.
– kolobokish
Dec 4 at 17:56
Sorry. I misspelled. It "can't" be finite and non-empty.
– kolobokish
Dec 4 at 17:56
add a comment |
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I can only say, that it can be finite and non-empty. In other words, it is either countably infinite or empty. (Both cases are possible, but it won't depend solely on choice of $delta$).
– kolobokish
Dec 4 at 17:02
Sorry. I misspelled. It "can't" be finite and non-empty.
– kolobokish
Dec 4 at 17:56