# Tag Info

### Who first proved ergodicity of irrational rotations of the circle?

The idea to study what we call irrational rotation of a torus indeed belongs to Nicole Oresme, at least he clearly understood the density of the trajectories (which is not the same as ergodicity or ...
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Accepted

### Simultaneous Diophantine approximation of $\sqrt{2}$ and $\sqrt{2\pm \sqrt{3}}$

$$\sqrt{2+\sqrt{3}}-\sqrt{2- \sqrt{3}}=\sqrt{2}$$
• 104k
Accepted

### Who first proved ergodicity of irrational rotations of the circle?

the proof goes back to Nicole Oresme in his paper De commensurabilitate vel incommensurabilitate motuum celi [On the Commensurability or Incommensurability of the Motions of the Heavens], dated around ...
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### For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

It's indeed unbounded for every irrational $x$. Let me identify points of $\mathbb{R}/\mathbb{Z}$ with their representatives on $[0,1)$, and order it by the usual order $<$ of $\mathbb{R}$ applied ...
• 6,437
Accepted

### Entropy of composition

There is a good reason you were having difficulties in proving this. This was an old question of Rohlin (MR0126526) which was first disproved in the topological setting by Goodwyn (MR0314023) and ...
Accepted

### Is the following series consisting of equally distributed $\pm 1$ bounded?

The sequence $\sum a_n$ is unbounded. This is a consequence of a general result from Kesten, On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arithmetica (1966). The ...
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Accepted

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### Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$

Here are a few simple remarks and one warning that are too long for a comment box. I surmise that you know most of them yourself but I'll just make them in case some reader finds any of them "non-...
• 60.1k
Accepted

### Ruelle-Perron-Frobenius theorem for shift of finite type

The most intuitive explanation I know is the following: suppose that you have a certain amount of mass (I usually picture a pile of sand) that is distributed over $\Sigma_A^+$ according to the density ...
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