Prove that the { fractional part of n α ∣ n ∈ N } is dense in [ 0 , 1 ] for an irrational number α .

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Prove that the { fractional part of   n  α  ∣  n  ∈      N   } is dense in   [  0  ,  1  ] for an irrational number     α  .

Harold Cantrell · Accepted Answer

Let us divide   [  0  ,  1  ] into   k intervals of length   1      /    k; i.e.   [  0  ,  1      /    k  ],   [  1      /    k  ,  2      /    k  ],   [  2      /    k  ,  3      /    k  ], etc.Now by Dirichlet principle there are two numbers   a  ≠  b such that   {  a  α  },   {  b  α  } which are in the same interval.If   b  &amp;gt;  a, then   (  b  −  a  ) is a positive integer and either   {  (  b  −  a  )  α  }  ∈  [  0  ,  1      /    k  ] or   {  (  b  −  a  )  α  }  ∈  [  1  −  1      /    k  ,  1  ].Since   α is irrational,   {  (  b  −  a  )  α  } is non-zero. (The number   (  b  −  a  )  α cannot be an integer.Now if we take all multiples   n  (  b  −  a  )  α,   n  ∈      N  , then in each of the k intervals must be at least one of the values   {  n  (  b  −  a  )  α  }. (We go either upwards from   [  0  ,  1      /    k  ], or downwards from the last interval, but we can never skip an interval.This implies that the set of all multiples is dense in   [  0  ,  1  ].

Prove that the { fractional part of n α ∣ n ∈<!-

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