Articles of equidistribution

Fractional part of $n\alpha$ is equidistributed

Let $\alpha$ be an irrational number. Then the sequence $\{\{n\alpha\}\}$ is equidistributed. I am using the following definition of equidistribution. A sequence $\{a_i\}$ is equidistributed if $\frac1n\sum\limits_{i=1}^n f(a_i) \to \lambda(f):=\int_0^1 f(x)\,dx$ for all continuous $f:[0,1]\to \mathbb{R}_{\ge 0}$. In the proof I used the fact that if $T_g(x)=(g+x) \pmod 1$, then $\lambda(f\circ T_g)=\lambda(f)$ for all continuous […]

When is a sequence $(x_n) \subset $ dense in $$?

Weyl’s criterion states that a sequence $(x_n) \subset [0,1]$ is equidistributed if and only if $$\lim_{n \to \infty} \frac{1}{n}\sum_{j = 0}^{n-1}e^{2\pi i \ell x_j} = 0$$ for ever non-zero integer $\ell$. I was wondering if anyone knows of a criterion similar to this one which characterizes when a sequence $(x_n) \subset [0,1]$ is dense in […]

Uniformly distributed rationals

Is there any algorithm, function or formula $f(n)$, which is a bijection between the positive integers and the rationals in $(0,1)$, with the condition, that for all real numbers $a,b,x$ with $0<a<b<1<x$, if we let $i(x)$ be the number of distinct integers $0<n_j<x$ which satisfy $a<f(n_j)<b$, then we have $\lim_{x\rightarrow\infty}i(x)/x=b-a$?

How to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in ?

$\forall \alpha\in [0,1]\setminus\mathbb{Q}$, how to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]? $\{x\}$ is the fractional part of x. Any hint would be appreciated!

Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?

Over in the thread “Evenly distributing n points on a sphere” this topic is touched upon: But what I would like to know is: “Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best. Does anyone know of a […]

$(n^2 \alpha \bmod 1)$ is equidistributed in $\mathbb{T}^2$ if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$

I did the following homework question, can you tell me if I have it right? We want to show that the sequence $(n^2 \alpha \bmod 1)$ is equidistributed if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. To that end we consider the transformation $T: (x,y) \mapsto (x + \alpha, y + 2x + \alpha)$ on the $2$-dimensional […]

Applications of equidistribution

What are applications of equidistributed sequences ? I’m looking for examples of problems/questions in fields (not necessarily mathematical) where equidistribution is an ad-hoc tool towards a solution.

Are the fractional parts of $\log \log n!$ equidistributed or dense in $$?

Are there any results relevant to the distribution of the sequence $\{\log \log n!\}$ for integers $n$, where $\{x\}$ denotes the fractional part of $x$? For instance, it is known that for irrational real numbers $\alpha$, the sequence $\{n\alpha\}$ is dense in $[0,1]$ and in fact equidistributed. Does something similar hold for the logarithms of […]

A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent

As the title says, I would like to launch a community project for proving that the series $$\sum_{n\geq 1}\frac{\sin(2^n)}{n}$$ is convergent. An extensive list of considerations follows. The first fact is that the inequality $$ \sum_{n=1}^{N}\sin(2^n)\ll N^{1-\varepsilon}\qquad\text{or}\qquad\sum_{n=1}^{N}e^{2^n i}\ll \frac{N}{\log(N)^{1+\varepsilon}} \tag{1}$$ for some $\varepsilon>0$ is enough to prove the claim by Abel summation. In the same […]

Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$

Suppose $f$ is continuous and periodic on the reals with period 1. Prove that if $x\in[0,1]$ is an irrational number, then $$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$$ Suggestion: First consider $f(t) = e^{2\pi(ikt)}$ where k is an integer. I can see that this is a limit of a weighted average, but the suggestion throws me off. I’ve seen the […]