# 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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## 1 Answer

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 $\{B(x,r):|x|<1/2\text{ and }|x|<r<(|x|+1)/3\}$.

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Whilst questions that are answered on Wikipedia pages are often considered a little weak for MO, I think that if it's answered on the discussion page then that constitutes "hard to find" and makes it okay! (Assuming that the level of mathematics is reasonable, which I'm not in a position to judge on this one.) –  Loop Space Dec 18 '09 at 12:20
I thought it was an interesting question, and it did take some work to find a counterexample. The level is hard for me to gauge; verifying that this is a counterexample is basic, but not everyone would quickly come up with it in the first place. I'll defer to others to judge whether this is reasonable. –  Jonas Meyer Dec 18 '09 at 23:27
I think this is interesting and very reasonable, just for the record. In fact, I'm a bit surprised this is in question. –  Ilya Grigoriev Feb 3 '10 at 6:14