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 ...
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 ...
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 ...
  • 5,136
26 votes
Accepted

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

$$\sqrt{2+\sqrt{3}}-\sqrt{2- \sqrt{3}}=\sqrt{2}$$
  • 90.3k
24 votes

Distribution of the Error term in GH Hardy's "curious result" $\sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$

There are several puzzling things about the question: Firstly of course $\theta$ must be irrational, and it is intended for $\{ x\}$ to denote the Bernoulli polynomial $x-[x]-1/2$ rather than the more ...
  • 42.5k
20 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 ...
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}^\...
16 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 ...
  • 17.1k
15 votes

Is there a generalized Birkhoff ergodic theorem?

The classical Birkhoff ergodic theorem considers a map $T$ acting on a probability space $(X,\mathcal{F},\mu)$. This could alternatively be thought of as an action of $\mathbb{N}$ on $(X,\mathcal{F},\...
  • 6,039
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 ...
  • 92.7k
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 ...
  • 15.6k
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 ...
  • 6,643
12 votes
Accepted

Closure of the orbits of the $SL(2,\mathbb{Z})$-action on $\mathbb{R}^2$

There are very precise results on the repartition of orbits of $SL(2,\mathbb{Z})$ of irrational points in $\mathbb{R}^2$, such as this one. In particular these orbits are dense, and this can be seen ...
  • 9,069
12 votes
Accepted

Possible limits of $(1/n) \sum_{k=0}^{n-1} e^{i2\pi \cdot 2^k\alpha}$

See the paper "Le poisson n'a pas d'arêtes" by Thierry Bousch. This is a joke that was explained to me much later. The set is considered to resemble a fish. The French word arête means both bone and ...
  • 21.7k
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 ...
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}...
12 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 $\...
  • 15.6k
12 votes

Is there a square with all corner points on the spiral $r=k\theta$, $0 \leq \theta \leq \infty$?

The points on the spiral with $\theta=$ $$ 3.4535999354657\\ 15.1248305526170\\ 22.0370015553781\\ 16.3950081067565 $$ form a square. The numbers can be generated from this Mathematica code, if ...
  • 17.7k
11 votes
Accepted

Who introduced the concept of topological mixing?

Topological mixing is called "permanent regional transitivity" (and demonstrated) by Gustav Hedlund in his 1939 article on the dynamics of the geodesic flow in constant negative curvature (third page)....
  • 17.1k
11 votes
Accepted

An algorithm for Poincare recurrence time

You want to find an $s$ such that $s, \sqrt{2} s, \sqrt{3} s, \sqrt{5} s$ are all close to integer. Your $t$ is then given by $2\pi s.$ The first question is a problem in simultaneous Diophantine ...
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 ...
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 ...
  • 15.6k
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....
10 votes
Accepted

Approximating Subshifts From Below

If $X$ is minimal and not a periodic orbit then it cannot contain a periodic orbit and hence in particular cannot contain a Markov shift. A classical construction by Grillenberger shows that one can ...
  • 6,039
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 ...
  • 21.7k
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 \...
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|\...
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 $z\mapsto\frac{1}{2}-\frac{1}{4}\left(z+\frac{1}{z}\right)$: $$ ...
  • 2,142

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