Density in [0; 1]

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  • Thread starter Calabi
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  • #1
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Hello, let be ##x \in \mathbb{R} - \mathbb{Q}##, do we have de density of ##\{nx - \lfloor{nx} / n \in \mathbb{N}\}## in ##[0; 1]## please?

I think yes but it's just an intuition : if I take a and b in ##[0; 1]## with a < b, I have an irrationnal between them let call it c but I don't know hat to do with it.

Could you help me please?

Thank you in advance and have a nice afternoon:oldbiggrin:.
 

Answers and Replies

  • #2
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In principle you already have said what you need. How is a dense subset formally defined? It might be a bit of work to formally prove that there are always irrational numbers as close to a given rational one as wanted. But given this fact, it's basically the definition of density itself.
E.g.: https://en.wikipedia.org/wiki/Dense_set
 
  • #3
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No that's not what I mean : the irrationnal number are dense in ##\mathbb{R}## so in ##[0; 1]## too (1)
And I wanted to use that to show that
##\{nx - \lfloor nx \rfloor / n \in \mathbb{N}\}## in [0; 1], where x is an irrationnal number, by using (1).
 
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