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Given an ergodic measure m on a shift space, by Shannon-Mcmillan-Breiman Theorem, up to at most an \epsilon-portion, $\epsilon$-portion, all cylinder sets of length n $n$ (large enough) have m-measure $m$-measure between exp(-nh-n\epsilon) $exp(-nh-n\epsilon)$ and exp(-nh+n\epsilon) $exp(-nh+n\epsilon)$ where h=h(m) $h=h(m)$ stands for the entropy. I wonder if there are other results making these lower and upper bounds tighter. Thank you!

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Given an ergodic measure m on a shift space, by Shannon-Mcmillan-Breiman Theorem, up to at most an \epsilon-portion, all cylinder sets of length n (large enough) have m-measure between exp(-nh-n\epsilon) and exp(-nh+n\epsilon)exp(-nh+n\epsilon) where h=h(m) stands for the entropy. I wonder if there are other results making these lower and upper bounds tighter. Thank you!

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Measure of large cylinder sets

Given an ergodic measure m on a shift space, by Shannon-Mcmillan-Breiman Theorem, up to at most an \epsilon-portion, all cylinder sets of length n (large enough) have m-measure between exp(-nh-n\epsilon) and exp(-nh+n\epsilon). I wonder if there are other results making these lower and upper bounds tighter. Thank you!