Questions tagged [equidistribution]

A bounded sequence of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that interval.

A bounded sequence $(s_n)$ of real numbers is equidistributed on an interval $[a,b]$ if for any subinterval $[c,d]$ of $[a,b]$ we have $\lim_{n\to\infty}{ \frac{|\{s_1,\dots,s_n \} \cap [c,d] |}n}={\frac{d-c}{b-a}}.$

Other related notions studied in the theory of uniform distributions are equidistribution modulo 1, discrepancy, well-distributed sequences, various kinds of distribution functions of sequences.

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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 $$…
Jack D'Aurizio
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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: https://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere. But what I would like to know is: "Is the Fibonacci lattice the very best…
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Show that $ \lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}e^{ik^2}=0$

TL;DR : The question is how do I show that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{ik^2}=0$ ? More generaly the question would be : given an increasing sequence of integers $(u_k)$ and an irrational number $\alpha$, how do I…
Renart
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Are the fractional parts of $\log \log n!$ equidistributed or dense in $[0,1]$?

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 $\{n\alpha\}$…
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$(n^2 \alpha \bmod 1)$ 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:…
Rudy the Reindeer
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When is a sequence $(x_n) \subset [0,1]$ dense in $[0,1]$?

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…
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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.
Gabriel Romon
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$\sin (n^2)$ diverges

One can prove that $\sin n$ diverges, using the fact that the natural numbers modulo $2\pi$ is dense. However, the case for $\sin (n^2)$ looks much more delicate since this is a subsequence of the former one. I strongly believe that this sequence is…
Yun
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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
TROLLHUNTER
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show that $ \limsup n\; | \;\{ (n+1)^2 \sqrt{2}\} - \{ n^2 \sqrt{2}\}\; | = \infty $

Here is a theorem from Kuipers-Neiderreiter: If $\{ x_n \}$ is a sequence uniformly distributed mod 1, then $\overline{\lim} n |x_{n+1} - x_n| = \infty$ I'm not 100% sure what this means so let's put an equidistibuted sequence $\{ n^2 \sqrt{2}\}$…
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Why does a golden angle based spiral produce evenly distributed points?

Vogel's model for sunflower seed arrangement uses discrete points on a spiral, with a very specific turning angle of $\theta_1 = 2\pi/\phi^2$ where $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio: $$ \theta_k = \frac{2\pi}{\phi^2} k, \qquad r_k =…
Spiros
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Equidistribution of $an^\sigma$ for $\sigma\in(0,1)$

I am stuck at an exercise in Stein's book: Fourier Analysis, and it's exercise 8 in chapter 4. Show that for any $a\ne0$, and $\sigma$ with $0<\sigma<1$, the sequence $an^\sigma$ is equidistributed in $[0,1)$. There are two…
gaoxinge
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Equidistribution of roots of prime cyclotomic polynomials to prime moduli

Here is a relevant - and longstanding, I'm told - conjecture. Let $f \in \mathbb{Z}[x]$ be irreducible and of degree > 1. Set $E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { normalised least positive residues of zeros of $f$ in…
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On the convergence of the series $\sum \frac{(-1)^{\lfloor ne\rfloor}}{n}$

The question is pretty simple: It the series $\sum \frac{(-1)^{\lfloor ne\rfloor}}{n}$ convergent or not ? As usual in this situation we let $S_n:=\sum_{k=0}^n(-1)^{\lfloor ne\rfloor}$ and apply an Abel transform to the original series. Thus…
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Prove that $\forall a\in \mathbb{R\smallsetminus Q}$, there exist infinitely many $n\in \mathbb N$ such that $\lfloor{an^2}\rfloor$ is even.

Question: Prove that $\forall a\in \mathbb{R\smallsetminus Q}$, there exist infinitely many $n\in \mathbb N$ such that $\lfloor{an^2}\rfloor$ is even. If $a=\sqrt2$ and $x^2-2y^2=1(x,y\in\mathbb N)$ then…
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