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hw4Bonus

hw4Bonus - Math508 Fall 2010 Jerry L Kazdan Bonus Problem...

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Math508, Fall 2010 Jerry L. Kazdan Bonus Problem for Set 4 1. Define two real numbers x and y to be equal if | x - y | is an integer. We write x y ( mod 1 ) . Thus we have a “topological circle” whose “circumference” is one. Let α be an irrational real number, 0 < α < 1 and consider its integer multiples, α , 2 α , 3 α ... ( mod 1 ) . Show that this set is dense in 0 x 1. S OLUTION Given any ε > 0 we’ll show that every point in 0 x 1 is ( mod 1 ) within ε of an integer multiple of α . Pick an integer K > 0 so that 1 / K < ε . Partition the interval [ 0 , 1 ] into the K intervals [ 0 , 1 / K ] , [ 1 / K , 1 / 2 K ] , ..., [( K - 1 ) / K , 1 ] , each of width 1 / K < ε . Consider the K + 1 (dis- tinct!) points α , 2 α ,... ( K + 1 ) α , ( mod 1 ) . Since there are K + 1 points and only
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