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Results tagged with ergodic-theory
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user 4312
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
8
votes
Accepted
Shannon entropy of $p(x)(1-p(x))$ is no less than entropy of $p(x)$
Denote $f(p)=p(1-p)$, $H(p)=-p\log p$, let $p_1,\dots,p_n$ denote all positive probabilities of our distribution, then $\sum p_i=1$, finally denote $s=\sum_i f(p_i)$. Then we need to prove the inequal …
- 109k
2
votes
Positive upper asymptotic density and equidistribution
No, take $B=\{k:\{k\alpha\}>\frac12 \}.$
- 109k
28
votes
Accepted
Simultaneous Diophantine approximation of $\sqrt{2}$ and $\sqrt{2\pm \sqrt{3}}$
$$\sqrt{2+\sqrt{3}}-\sqrt{2- \sqrt{3}}=\sqrt{2}$$
- 109k