Timeline for Precise density estimates for Cantor sets
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 25, 2014 at 21:39 | vote | accept | Paul-Benjamin | ||
| Nov 24, 2014 at 22:42 | comment | added | Pablo Shmerkin | @Asaf - this has very little to do with Shannon-McMillan-Breiman or ergodic theory in general. Besides giving only a.e. results, S-M-B or the ergodic theorem give information about local dimension (i.e. asymptotic behavior of $\log \mu(B(x,r))/\log r$), not about densities (i.e. asymptotic behavior of $\mu(B(x,r))/(2r)^s$). But for this simple construction one can still find sharp bounds valid at every point by direct geometric considerations. | |
| Nov 24, 2014 at 22:38 | history | edited | Pablo Shmerkin | CC BY-SA 3.0 |
added 1230 characters in body
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| Nov 24, 2014 at 16:02 | comment | added | Asaf | I think that in-general, using the Shannon-McMillan-Breiman theorem, one can get a bound for "most points", although a bound for every point is much more delicate, and I don't think it there will be one in general (as the exceptional sets for the ergodic theorem in a Bernoulli system is rather large). | |
| S Nov 24, 2014 at 11:07 | history | suggested | Paul-Benjamin | CC BY-SA 3.0 |
Intersection with C
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| Nov 24, 2014 at 10:44 | comment | added | Paul-Benjamin | Thank you very much for your answer Mr. Shmerkin. But there is still something's missing. In the exercise, I had to determine bounds $0<a<b<c<d<\infty$ such that $a\leq\Theta_*\leq b<c\leq \Theta^*\leq d$. As far as I can see, we can take $a=2^{(-s+1)},c=2^{-s},d=1$, but I don't know how to get $b$ such that $a<b<c$, because in the article you mentionned, there is an upper bound for the lower density almost everywhere, but I'm supposed to get an estimate everywhere. | |
| Nov 24, 2014 at 10:29 | review | Suggested edits | |||
| S Nov 24, 2014 at 11:07 | |||||
| Nov 24, 2014 at 1:24 | history | answered | Pablo Shmerkin | CC BY-SA 3.0 |