# Why is 3 a bad constant in the Vitali covering lemma?

Hi,

recently I had to do with the Hardy-Littlewood maximal function and we used there the Vitali covering lemma with constant 5. Then, given an advice, I proved it with constant k>3. But I cannot find an counterexapmle why 3 is not enough (it is enough in the finite version of the lemma). Has anyone seen such an example?

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An example in $\mathbb{R}$ was posted by Oded Schramm in May 2008 on the discussion page of the Wikipedia article on the Vitali covering lemma. Namely, consider $\{B(x,r):|x|<1/2\text{ and }|x|<r<(|x|+1)/3\}$.