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Does there exist a bijection $f: \mathbf{N} \to \mathbf{N}$ such that $$ \forall A \subseteq \mathbf{N}, \quad \sum_{n \in A}\frac{1}{n}<\infty \Longleftrightarrow \sum_{n \in A}\frac{1}{\sqrt{f(n)}}<\infty\,\,? $$

Paolo Leonetti
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  • Well $\implies$ is easy since you can just define $f(n)=n$, but as for the $\impliedby$ case, not so easy. – Ty Jensen Oct 27 '20 at 17:59
  • I believe the answer is no. But I don't have a proof, just a lack of examples. – Paolo Leonetti Oct 27 '20 at 18:00
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    Very good question by the way ! (+1) :-) – Surb Oct 27 '20 at 18:01
  • @ Ty Jensen don't you mean the other way round? – Adam Rubinson Oct 27 '20 at 18:02
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    Does this answer the question: https://math.stackexchange.com/questions/2429389/is-there-a-bijection-of-the-natural-numbers-which-swaps-frac1n-summable-s?rq=1 – Adam Rubinson Oct 27 '20 at 18:15
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    @AdamRubinson Surely yes! Thank you Adam. Please vote to close this question – Paolo Leonetti Oct 27 '20 at 18:16
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    Thank you for the fun question Paolo! – Ty Jensen Oct 27 '20 at 18:17
  • @TyJensen You are welcome, I am just afraid it was a duplicate, I didn't know it – Paolo Leonetti Oct 27 '20 at 18:18
  • I'm not sure if "voting to close" it the same as "flagging as duplicate", but I don't know how to do the former, so I've done the latter. – Adam Rubinson Oct 27 '20 at 18:18
  • Paolo at least you don't have to give away 500 reputation to get an answer like the original questioner did! To be fair, that answer is probably worth 500 rep anyway. – Adam Rubinson Oct 27 '20 at 18:20
  • @AdamRubinson Ahaha this should be a classical result for Rudin--Keisler isomorphic (summable) ideals of N, but I just didn't figure out how to prove it; that's why I posted it here :) Should I delete completely the question? – Paolo Leonetti Oct 27 '20 at 18:22
  • Not sure. See: https://math.meta.stackexchange.com/questions/29191/should-duplicate-questions-be-deleted and decide for yourself – Adam Rubinson Oct 27 '20 at 18:25
  • @AdamRubinson Thanks for the link: basically, it states that "we should leave it here iff it is useful to avoid further duplicates". I believe that the title is sufficiently different, even if at first sight we discover it is exactly the same problem. If someone else thinks differently, let me know and I will delete it. – Paolo Leonetti Oct 27 '20 at 18:29
  • This will be of interest for a future reader: https://arxiv.org/pdf/1809.04658.pdf – Paolo Leonetti Oct 27 '20 at 18:50
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    I agree that your phrasing of the question is significantly different and more readable than the original question, so I agree, don’t delete it – Adam Rubinson Oct 27 '20 at 18:56

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