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Math Help - show a set has measure 0

  1. #1
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    show a set has measure 0

    Let \Omega denote the set of numbers x \in [0,1] such that there exist infinitely many
    rationals \frac{p}{q}, with p and q coprime, with the property
    |x-\frac{p}{q}|<\frac{1}{q^3}. Show that \Omega has measure 0.

    I think I should use the Borel Cantelli Lemma for this, however I am struggling to think of a suitable set An so that limsupAn= \Omega. Can you help me try to find a suitable set?
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  2. #2
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    Quote Originally Posted by ramdayal9 View Post
    Let \Omega denote the set of numbers x \in [0,1] such that there exist infinitely many
    rationals \frac{p}{q}, with p and q coprime, with the property
    |x-\frac{p}{q}|<\frac{1}{q^3}. Show that \Omega has measure 0.

    I think I should use the Borel Cantelli Lemma for this, however I am struggling to think of a suitable set An so that limsupAn= \Omega. Can you help me try to find a suitable set?
    It seems to work if you let A_q=\{x\in[0,1] : \left|x-\frac pq\right|<\frac{1}{q^3} for some 0\leq p<q\}. This excludes the rationals \frac{p}{q}\notin[0,1] but there are only finitely many that fit if x\in(0,1), I think, so this doesn't matter (to be checked). Alternatively, you can allow 0\leq p<2q in the definition to settle this problem more easily (if |x-p/q|<1/q^3 then p/q\leq |x|+1/q^3\leq 1+1=2)
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