show a set has measure 0

**ramdayal9** · April 20th 2010, 01:49 PM

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?

**Laurent** · April 20th 2010, 02:06 PM

Originally Posted by ramdayal9

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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