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Let $0 \leq \alpha < \beta \leq 1$. I'm looking for an example of a Lebesgue measurable subset $E$ of $\mathbb{R}$ such that

$\liminf_{\delta \rightarrow 0} \frac{m(E \cap (-\delta,\delta))}{2\delta} = \alpha$

but

$\limsup_{\delta \rightarrow 0} \frac{m(E \cap (-\delta,\delta))}{2\delta} = \beta$

where $m$ is the Lebesgue measure on $\mathbb{R}$.

Can someone give an example? Thank you, Malik

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    (I deleted 2 comme$n$ts that were posted on the merged question that no longer made sense. The comment of Yuval Filmus that presently precedes this one was originally posted on the merged question. http://math.stackexchange.com/questions/28577/sets-with-prescribed-upper-and-lower-densities)2011-03-23

3 Answers 3

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Try the duplication across zero of $\bigcup_{n \geq 1} \left[\frac{1}{(2n)!}-\alpha\left(\frac{1}{(2n)!} - \frac{1}{(2n+1)!}\right),\frac{1}{(2n)!}\right] \cup \left[\frac{1}{(2n+1)!}-\beta\left(\frac{1}{(2n+1)!} - \frac{1}{(2n+2)!}\right),\frac{1}{(2n+1)!}\right].$

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    @YuvalFilmus In case you have the time, see this question: http://math.stackexchange.com/questions/2063959/construct-a-set-with-different-upper-and-lower-lebesgue-density-at-zero2016-12-18
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Divide the interval $(0,1]$ of radii into intervals which decrease "fast enough". In some of the intervals put some set of density $\alpha$, and in other put some set of density $\beta$. You can fill-in the rest of the details yourself.

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I think the following construction works: Given $\alpha \leq \beta$, define sequences $(a_n)_n$ and $(b_n)_n$ as follows: $a_0 = 0;$ $b_0 = 1;$ $a_n = (\beta/\alpha)b_{n-1};$ $b_n = {1-\alpha \over 1-\beta}a_n.$ Now define $E_n = [a_n, b_n)$ and $E = \bigcup_{n=0}^\infty E_n$.

The sequences were chosen so that $\bigcup_{i=0}^n E_i$ contains (roughly) $\beta$ of $[0, b_n)$ but only $\alpha$ of $[0, a_{n+1})$. It actually always contains slightly more than this, because it contains all of $[0, 1)$ instead of some crazy fractal pattern inside it, but any finite initial segment doesn't matter to the problem. As $R \rightarrow \infty$, the density of $E \cap [0, R)$ in $[0, R)$ oscillates (linearly!) between the two values, giving the behavior you requested.

But don't take this as gospel. It's been a while since I did any measure theory and I wasn't much good at it even at the time.