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46 votes

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 ...
Alexandre Eremenko's user avatar
28 votes
Accepted

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

$$\sqrt{2+\sqrt{3}}-\sqrt{2- \sqrt{3}}=\sqrt{2}$$
Fedor Petrov's user avatar
28 votes
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 ...
Carlo Beenakker's user avatar
27 votes

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 ...
Ville Salo's user avatar
  • 6,437
21 votes
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 ...
Nishant Chandgotia's user avatar
17 votes
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 ...
coudy's user avatar
  • 18.6k
17 votes
Accepted

Almost all non-negative real numbers have only finitely many multiples lying in a measurable set with finite measure

Let $f(t) = 1$ if $t \in A$ and $f(t) = 0$ otherwise. Suppose that $a > 0$. Then $$ \begin{aligned} \int_a^{2 a} \operatorname{card} \{n : n t \in A\} dt & = \int_a^\infty \biggl(\sum_{n = 1}^\...
Mateusz Kwaśnicki's user avatar
16 votes
Accepted

If the pointwise ergodic theorem holds along all subsequences with nonzero natural density, is the system strong mixing?

One can construct a counterexample using a stationary Gaussian process. The basic idea is to use a system that is weak mixing with very good bounds, without being strong mixing, as it is the former ...
Terry Tao's user avatar
  • 110k
15 votes
Accepted

Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?

Where to begin! The ergodicity of non-compact subgroups (singular tori) was used by Margulis to prove that higher rank lattices $\Gamma$ are arithmetic. Once you have that $\Gamma $ is arithmetic, ...
14 votes
Accepted

Are there numbers whose binary and ternary representations simultaneously have few digit transitions? How frequent are those numbers?

[Edited because I had misread $c_b(k)$ to be the number of non-zero digits in the base $b$ representation, rather than the number of digit transitions. The argument works for both variants. -T] The ...
Terry Tao's user avatar
  • 110k
13 votes
Accepted

A kind of converse to the Hopf theorem on ergodicity of geodesic flow in negative curvature

It was proved by Donnay that any compact orientable surface can be given a Riemannian metric for which the geodesic flow is ergodic. See Theorem 1 of this article. On the other hand, there are no ...
R W's user avatar
  • 16.7k
13 votes

Uniform distribution of points on Riemannian manifolds

Let $\mu=(\delta_A+\delta_B)/2$. Then the claim of Arnold - Krylov is the weak convergence of the convolutions $\mu^{*n}*\delta_x$ to the rotation invariant probability measure on the sphere (where $\...
R W's user avatar
  • 16.7k
13 votes

For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

We have Theorem. Let $\psi(x)$ and $\varphi(x)$ be positive increasing functions such that $$\int_1^\infty \frac{dx}{\psi(x)}=+\infty,\qquad \int_1^\infty \frac{dx}{\varphi(x)}<+\infty.$$ Then ...
juan's user avatar
  • 6,974
12 votes

About positive upper density

Suppose that $(S-n)\cap S$ has upper density 0 for all $n\leq N$. Then the density of the set of integers $x$, such that $|[xN, (x+1)N]\cap S|\geq 2$ is also 0, hence the upper density of $S$ is $\leq ...
Jan-Christoph Schlage-Puchta's user avatar
12 votes
Accepted

Is there a complete Riemannian manifold with infinite volume whose the time-one map of the geodesic flow is recurrent?

Take a compact, connected Riemannian manifold $M$ with negative sectional curvature. Then choose any cover $M'$ of $M$ which is connected, Galois, and whose group of deck transformations is $\mathbb{Z}...
D. Thomine's user avatar
11 votes
Accepted

Conditional convergence of $\sum_{n\geq 1} \frac{\sin(p(n))}{n}$?

This is something I learned from fedja (artofproblemsolving.com): We will show that if $x$ is not Liouvillian then the sum $$ \sum_{n \geq 1} \frac{\sin (2 \pi x n^{2})}{n} $$ converges. An ...
Paata Ivanishvili's user avatar
11 votes
Accepted

A question about billiards

The answer is given by the general theory of rational billiard flows (i.e., those on polygons whose angles are rational multiples of $\pi$). On such a polygon $Q$ the tangent vectors to any given ...
R W's user avatar
  • 16.7k
11 votes
Accepted

Density-$c_0$ in $\ell^\infty$

This type of convergence is often called statistical convergence. The paper Constantin P. Niculescu, Gabriel T. Prajitura: Some open problems concerning the convergence of positive series (arXiv:1201....
Martin Sleziak's user avatar
11 votes
Accepted

If the average of a sequence converges, can I find a uniform bound that does not depend on where I start?

The sequence $1,0,1,1,0,0,1,1,1,0,0,0,\ldots$ is a counterexample. For each $j$ we have $\frac{1}{n}\sum_{k=1}^n a_{k+j} \to \frac{1}{2}$, but for any proposed $N_\epsilon$ we can find a value of $j$ ...
Nik Weaver's user avatar
  • 42.4k
10 votes

Solve the functional equation $f(4x(1-x))=\sin(\pi f(x))$ to find an invariant measure of a dynamical system $x_{n+1}=\sin(\pi x_{n})$

What you are asking for is a conjugacy of the dynamical systems $g(x)=4x(1-x)$ and $h(x)=\sin(\pi x)$. Since $g$ and $h$ are both full unimodal maps of $[0,1]$, there will exist such a conjugacy, but ...
Anthony Quas's user avatar
  • 22.5k
10 votes
Accepted

The mean value of $y \log{y}$ over the ordinates of the CM points

Let $g : \Gamma \backslash \mathbb{H} \to \mathbb{C}$ be any bounded continuous function. Duke's theorem states that \[\frac{1}{h(D)} \sum_{A \in \mathrm{Cl}_K} g(z_A) = \frac{1}{\mathrm{vol}(\Gamma \...
Peter Humphries's user avatar
10 votes

For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

Here is an argument essentially due to fedja I learned about thirteen years ago on artofproblemsolving.com. Proposition: if $f$ is $2$-periodic Riemann integrable such that $\sup_{n \geq 1} \left|\...
Paata Ivanishvili's user avatar
10 votes
Accepted

Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces

The answer is yes. First, it follows from the following result and the Riesz–Markov–Kakutani representation theorem that we can always find a suitable Baire measure representing a positive linear ...
Michael Greinecker's user avatar
10 votes
Accepted

Ergodic measures for the logistic map

When $c=4$, the map $T_4(x)=4x(1-x)$ on the unit interval is semi-conjugate to the transformation $z\mapsto z^2$ of the unit circle via $z\mapsto\frac{1}{2}-\frac{1}{4}\left(z+\frac{1}{z}\right)$: $$ ...
KhashF's user avatar
  • 3,230
10 votes

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-...
fedja's user avatar
  • 60.1k
9 votes
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 ...
Vaughn Climenhaga's user avatar
9 votes

All two-point correlations equal to $0$, three-point correlation not $0$?

I think you could make such an example by choosing any normal sequence $S$ on the alphabet $\{0,1,2,3\}$, and then applying the letter-to-word substitution $\tau$ defined by $0 \mapsto +++$, $1 \...
Ronnie Pavlov's user avatar
9 votes
Accepted

Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?

There is a simple reduction of the Birkhoff ergodic Theorem for $L^1$ functions to the bounded case using Kakutani-Rokhlin towers, that I learned from H. Furstenberg and B. Weiss decades ago. We use ...
Yuval Peres's user avatar
9 votes
Accepted

Let $(a_n)_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the set$\{2^n3^m\mid m,n\in N\}$. Find $α$ such that $(a_n)\alpha\pmod1$ is not equidistributed

$\alpha =\sum_{n=1}^{\infty} \frac{1}{ 6^{100^n}}$ should do the trick. A positive proportion of numbers on your list are of the form $2^a 3^b$ for $a,b$ within a reasonable constant factor of each ...
Will Sawin's user avatar
  • 138k

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