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I read from the book "Which Way did the Bicycle Go" that it is unknown whether for every $c>0$ there are infinitely many integers $n$ such that $|n\sin n|<c.$

Let $\mathbb{Q}_{m}$ be the set of rationals where the absolute value of the denominator is at most $m$ when the rational number has been written in its simplest form.

What can we say about the following weakened problem?

For every $c>0$ there is an integer $m$ and infinitely many rationals $q\in\mathbb{Q}_m$ such that $|q\sin q|<c.$

Has this been solved for some fixed $m?$

AlexR
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mathprogrammer
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  • How could you solve it for fixed $n$? That does not really make sense given the statement. – Tobias Kildetoft Dec 12 '13 at 12:24
  • Your edit to change $n$ to $m$ does not change the sense it makes. – Tobias Kildetoft Dec 12 '13 at 12:35
  • Yes. I used the letter $n$ in two meanings. What I tried to write was to allow denomimator to be some number not too large integer. I changed the problem. – mathprogrammer Dec 12 '13 at 12:37
  • You are still asking for the existence of $m$, which would not make sense if you fix $m$. – Tobias Kildetoft Dec 12 '13 at 12:38
  • What I tried to say was that the original problem ask whether there are infinitely many integers. What I meant was that if we take all integers plus rationals of the form $1/2,2/2,\ldots,1/3,2/3,\ldots,a/(m-1)$, denote it say by $\mathbb{Q}{restricted}$ and ask if one can find infinitely many of those numbers such that $|q\sin q|<c$ for infinitely many $q\in\mathbb{Q}{restricted}.$ Yeah, I have some problems to formalize my the problem I have on my mind. Someone might edit the problem. – mathprogrammer Dec 12 '13 at 12:46

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