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Finding an Irrational not in the Union.

  1. Sep 1, 2011 #1
    Hi, All:

    This is an old problem I never solved, and I recently saw somewhere else:

    We are given an enumeration {q_1,q_2,..,q_n,....} of the rationals in the real line.

    We construct the union : S:=\/ (q_i+[e/2^(i+1)] , q_i-[e/2^(i+1)] ) for i=1,2..,n,..

    i.e., we want the total measure of S to be a real eps>0; a fixed real number.

    Question: given a choice of enumeration, we want to find an irrational that is _not_

    in S. I think the best idea is using Roth-Siegel approximation and somehow

    using an algebraic number that cannot be approximated too well by rationals with

    fixed denominator, but I can't think of a full solution. Any Ideas?

    number that cannot
  2. jcsd
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