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I'm thinking of the set $S_r = \{nr \mod 1|n\in \mathbb{Z^+}\}$ for a fixed irrational number $r\in[0,1)$.

I just found that from an algebraic perspective, $S_r$ is equivalent to the set generated by $r$ in $\mathbb{R}/\mathbb{Z}$ (or may thinking about unit circle group).

There are many good questions about the set of rational numbers in [0,1) (even it forms a group), but I cannot find any good properties about irrational numbers for $S_r$.

I noticed the following facts:

  1. $S_r$ is dense.
  2. $q \notin S_r$ where $q$ is a rational number.
  3. If r is an algebraic number, every transcendental number in $[0,1)$ is not in $S_r$.
  4. If r is a transcendental number, every algebraic number in $[0,1)$ is not in $S_r$.

But still cannot cover all the numbers. Is there a named set or known keywords about $S_r$?

NeutrinoAnt
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  • @Joe Sorry for that, I edited it. – NeutrinoAnt Mar 22 '21 at 04:55
  • related: https://math.stackexchange.com/questions/843763/for-x-in-mathbb-r-setminus-mathbb-q-the-set-nx-lfloor-nx-rfloor-n-in-ma – Joe Mar 22 '21 at 13:50
  • https://math.stackexchange.com/questions/450493/positive-integer-multiples-of-an-irrational-mod-1-are-dense/450530 – Joe Mar 22 '21 at 13:50
  • https://math.stackexchange.com/questions/903142/for-an-irrational-number-a-the-fractional-part-of-na-for-n-in-mathbb-n-is – Joe Mar 22 '21 at 13:50
  • https://math.stackexchange.com/questions/3337395/density-of-a-set-multiples-of-irrational-number-mod-irrational-number – Joe Mar 22 '21 at 13:51
  • @Joe Thanks for the links. I think the links are related to just a 'density', not an element in it, but I just realized $S_r$ has only elements of the form $a+br$ where $a,b\in\mathbb{Z}$ and $b\gt 0$, trivial.. so I wrapped up. – NeutrinoAnt Mar 23 '21 at 01:50

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