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rigliztetbf

rigliztetbf

Answered question

2022-06-05

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

Answer & Explanation

Harold Cantrell

Harold Cantrell

Beginner2022-06-06Added 21 answers

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 > 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 ].

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