The set { n 2 mod 1 : n ∈ N } is dense in ( 0 , 1 ), it follows from the irrationality of 2 (a well-known fact, that fractional parts of multiples of irrational number form a dense in ( 0 , 1 )).Is { n 2 mod 3 : n ∈ N } dense in ( 0 , 3 ) ?Is { n 2 mod π : n ∈ N } dense in ( 0 , π ) ?Probably the answer is yes. Can it be derived from the above, or this is a completely different result requiring much more advanced theory?

Question

The set   {  n      2    mod  1  :  n  ∈      N    } is dense in   (  0  ,  1  ), it follows from the irrationality of       2   (a well-known fact, that fractional parts of multiples of irrational number form a dense in   (  0  ,  1  )).Is   {  n      2    mod      3    :  n  ∈      N    } dense in   (  0  ,      3    ) ?Is   {  n      2    mod  π  :  n  ∈      N    } dense in   (  0  ,  π  ) ?Probably the answer is yes. Can it be derived from the above, or this is a completely different result requiring much more advanced theory?

Shawn Castaneda · Accepted Answer

In order for   {  n  a    mod    b  :  n  ∈      N    } to be dense in   (  0  ,  b  ), it is necessary and sufficient that   a      /    b be irrational. This is true for   b  =  1, as your question seems to acknowledge. The general case can be derived from the special case   b  =  1 by using the transformation  f  :  (  0  ,  1  )  →  (  0  ,  b  ) defined by   f  (  t  )  =  b  t.

The set <mo fence="false" stretchy="false">{ n <msqrt> 2 </msqrt> <mo lspace="t

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