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.
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Probability that the first $m$ digits in $2^n$ are $k_1k_2\dots k_m$
I want to find the probability that the first $m$ digits of powers of 2 are a given combination $k_1k_2\dots k_m$. So far, here's my reasoning:
A number $2^n$ will have the first $m$ digits of the ...
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When is the convolution of two sequences equidistributed?
A sequence of integers $(x_n)$ is equidistributed mod $p$ if for all $a$ mod $p$, we have as $n \to \infty$:
$$ \dfrac{1}{X} \# \{n < X: x_n \equiv a \mod p \} \to \dfrac{1}{p}.$$
Let $(a_n)$ and $...
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How does this algorithm for the Van der Corput sequence work?
For any natural number $n$ write its binary expansion as $n = \sum_{i=0}^{k(n)} n_i 2^i$. Then the $n$th entry of the binary Van der Corput sequence is defined to be the dyadic rational
$$V(n) = \sum_{...
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If $x > 1$ then does the set $ \lbrace{ \lbrace{ x^n\rbrace}: n\in\mathbb{N} \rbrace}\ $ always contain either $0$ or $1$ as a limit point?
This question is somewhat related to my previous question here.
Here, $\lbrace{ \cdot \rbrace}$ means fractional part.
Is it true that if $x > 1\ $ then either $0$ or $1$ (or both) are accumulation ...
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Equidistributed Sequences and Riemann-Integrability
Recently I have been researching about equidistributed sequences. In the Wikipedia page: https://en.wikipedia.org/wiki/Equidistributed_sequence, I found the next theorem:
The sequence $\{s_n\}_{n=1}^\...
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distribution of toric orbits in $\mathrm{SL}_2(\mathbb{Q}_p)$
Write $G=\mathrm{SL}_2(\mathbb{Q}_p)$ where $p$ is a prime number, $\Gamma$ a cocompact subgroup of $G$ and $T$ the subgroup of $G$ consisting of diagonal matrices of $G$. Is the product $T\cdot\Gamma$...
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Equidistribution of set of numbers and moments
Equidistribution of a set of numbers $\{s_1, s_2, s_3, \cdots\}$, loosely speaking, is the property that the proportion of terms falling in a subinterval is proportional to the length of that ...
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For any $\epsilon>0$, there exists arbitrarily large $x$ with $\cos( x^2)>1-\epsilon$ and $\cos[ (x+1)^2]<-1+\epsilon$
For any $\epsilon>0$, there exists arbitrarily large $x$ with $\cos (x^2)>1-\epsilon$ and $\cos [(x+1)^2]<-1+\epsilon$.
This is an exercise in "Uniform Distribution of Sequences" ...
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If $\lim_{x\to \infty} f^{(k)}(x)=\theta$ (irrational), then $(f(n))$ is uniformly distributed modulo $1$
Let $f\in \mathcal C^{k+1}$, $x\ge 1$ and let for some integer $k\ge 1$, we have
$$\lim_{x\to \infty} f^{(k)}(x)=\theta$$
for irrational $\theta$. Then, prove that $(f(n))$ is uniformly distributed ...
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Does such an "extremely" equally distributed real, bounded sequence exist?
For each $\ n\in\mathbb{N},\ $ define the set $\ [n] := \{1,2,\ldots, n\}.\ $ Does there exist a real sequence $\ (a_n)_{n\in\mathbb{N}}\subset [0,1),\ $ such that for each $\ n\in\mathbb{N}\ $ there ...
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fractional part of $n^2\alpha$ is equidistributed
Question:
$\alpha$ is an irrational number.Denoted the fractional part of $n^2\alpha$ as $\{n^2\alpha\}$.Prove that $\{n^2\alpha\}$ is equidistributed,i.e.
$$\lim_\limits{N\to\infty} \frac{\#\{1\leq n\...
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Is there a sequence so that $\sum |a_n|=\infty$ and $\sum a_n \cos(nx)$ and $\sum a_n \sin(nx)$ converge everywhere?
Let the series be $(s_n a_n)_n$ where $s_n\in\{-1,1\}$ and $a_n$ decreases to $0$ instead for convenience. If $s_n$ is eventually periodic you can choose an $x$ which is a rational multiple of $\pi$ ...
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Ref proof for equidistribution of roots of $x^2 \equiv 1$ (mod $p$), as $p$ varies over primes?
This question is about equidistribution of roots of quadratic congruences of the form $x^2 \equiv n$ (mod $p$), which we note always have determinant $D = b^2 - 4ac = 4n$. This is for a fixed $n$ ...
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Durrett's Probability: Theorem 6.2.6
I am having some difficulty understanding the concept of measure preserving, invariance, and ergodic.
Here is a proof from Theorem 6.2.6 in Durrett's Probability: Theory and Examples, 5e (p.338) (...
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How do we show such set is equidistributed when it is involved from the image of local set?
Let $a \in \mathbb{R}\backslash\mathbb{Q}$ be an generic irrational number. Then, it's well-known that $\{ n a \text{ mod } 1\mid n\in \mathbb{N}\}$ is equidistributed on [0, 1).
My question is,
...
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Inequality for sum of sines over integers in $[1;x]$
It's easy to prove that $\sum_\limits{n=1}^x|\sin(n)|\sim\frac{2x}{\pi}$, using equdistribution of $\{n\;(mod\;\pi)\}$ on $[0;1]$. Define $S(x)=\sum_\limits{n=1}^x\left(|\sin(n)|-\frac{2}{\pi}\right)$....
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Equidistribution of Polynomials with At Least One Irrational Coefficients
This is Problem 2 from Stein and Shakarchi's Fourier Analysis, Chapter 4. Below is the problem.
Here we present an estimate of Weyl which leads to some interesting results.
(a) Let $S_N = \sum_{n=1}^{...
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Show that $\lim_{N\to\infty}\int_0^1 \left|\frac1N\sum_{n=1}^N f(x+\xi_n) \right|^{\,2}\, dx = 0$ if $\int_0^1 f(x)\, dx = 0$ and $f$ is periodic
Suppose $f$ is a periodic function on $\mathbb R$ of period $1$, and $\{\xi_n\}_{n=1}^\infty$ is a sequence which is equidistributed in $[0,1)$. Prove that if $f$ is Riemann integrable on $[0,1]$ and ...
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What is wrong with the following proof that $\{ (\frac{3}{2})^n\mod 1: n\in\mathbb{N} \} $ is dense in $\ [0,1]\ $?
What is wrong with the following straightforward proof that $\ \lbrace{ \left(\frac{3}{2}\right)^n \mod 1: n\in\mathbb{N} \rbrace}\ $ is dense in $\ [0,1]\ $ ?
In fact, let $\ f(x) = \lbrace{\ x^n \...
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Does equidistribution of $(a_n)_{n\in\mathbb{N}}$ imply equidistribution of $(a_n+a_{n+k})_{n\in\mathbb{N}}$?
Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers that is equidistributed modulo 1 and let $k\in\mathbb{N}$.
Then it is clear that the sequence $(a_{n+k})_{n\in\mathbb{N}}$ is also ...
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Understanding the Single-scale equidistribution theorem for abelian polynomial sequences using example
I wish to understand a remark given in the book 'Higher order fourier analysis' by T. Tao. The remark is related to the following proposition:
Proposition 1.1.17 (Single-scale equidistribution theorem ...
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Ratner's theorem on equidistribution for abelian linear sequences
This question is a doubt arising from the statement and the proof of proposition described below ( Book: Higher Order fourier analysis, by T. Tao)
Suppose $\alpha \in \mathbf{T}^d$. Then,
Irrational :$...
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Something similar to Kronecker's Theorem
I would like to prove (or have a reference to) the following:
Given $n$ real numbers $a_1,\ldots,a_n>0$ and $\varepsilon>0$, there exist $k,k_1,\ldots,k_n\in\mathbb N\setminus\{0\}$ such that $|...
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Uniform distribution of points on a sphere
I'm reading paper by Arnol'd and Krylov ( UNIFORM DISTRIBUTION OF POINTS ON A SPHERE AND SOME ERGODIC PROPERTIES OF SOLUTIONS OF LINEAR ORDINARY DIFFERENTIAL EQUATIONS IN A COMPLEX REGION ).
I need ...
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Effective equidistribution for number with bounded irrationality measure
This question asked for bounds greater than $\omicron\left(n\right)$ on the error
$$
E_n=|T\cap\{1,2,\ldots\}|-\ell n.
$$
where
$$
\ell=\lim_{n\to\infty}\frac{|T\cap\{1,2,\ldots\}|}{n} \qquad\text{ ...
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Fractional part of a log n is not equidistributed
In Exercise 9 in Fourier Analysis, Stein, I want to prove that fractional part of $a \log n$ is not equidistributed for any $a$.
Using Weyl's criterion, my attempt is to show that $$\frac1N\sum_{n=1}^...
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Equidistribution theorem in two dimension
If $\alpha\in \mathbb R$ is an irrational, then $\{ \langle k\alpha \rangle \}_{k \ge 1}$ is dense in $[0,1]$, where $\langle x \rangle$ denote the fractional part of $x$. Moreover, $\{ \langle k\...
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${α⋅ \log(n)}$ is not uniformly distributed mod1 in $[0,1]$
$\qquad \qquad \bbox[15px,border:2px solid red] { x_n:=\text{\{α$\cdot$ log(n)\}}_{n\in \mathbb N}}$
I want to show that the sequence $x_n$ is not uniformly distributed mod1 in $[0, 1]$ for any $α\in ...
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Condition for ({$n\alpha$},{$n\beta$}) being dense on [0,1]$\times$[0,1] [duplicate]
What is the condition for ({$n\alpha$},{$n\beta$}) being dense in [0,1]$\times$[0,1]? (Here {a} represents the fraction part of a).
If the qustion changes to ({$n\alpha$},{$m\beta$}), then it is ...
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Equidistribution of powers of primitive roots modulo $p$
Let me start with a nice experimental observation. Fix a large prime, say $p = 5003$. It turns out that $g = 2$ is a primitive root mod $p$. If we plot the powers of $g \in \Bbb F_p^{\times}$ (...
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General Case of Weyl's Equidistribution Theorem
Weyl's Theorem says that if $p(x)$ is a polynomial with at least one of the coefficients (non-constant) is irrational then the sequence $\{p(n)\}$ is equidistributed in $\mathbb T$ (Torus of $1$ ...
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Distribution of $\{n^p\alpha\}$ for irrational $\alpha$
Let $\alpha$ be an irational number. Consider sequence $x_n=\{n^p\alpha\}$, $n\in\mathbb{N}$ (it's the fractional part of $n^p\alpha$), where $p$ is a nonzero real number.
Question. For which values ...
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On the Equidistribution Weyl Theorem for $\{2^kx\}$
It is well known that the sequence $\{2^k \eta\} \bmod 1$ is uniformly distributed for almost all, but not all, irrational $\eta$ in $(0,1)$. If I fix an irrational number $0<x<1$ ($x$ is ...
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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 ...
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Show that the sequence $\frac{0}{1}, \frac{0}{2}, \frac{1}{2}, \frac{0}{3}, \frac{1}{3}, \frac{2}{3}, \dots \frac{k-1}{k}$ is equidistributed mod 1
I need to show that the sequence $\frac{0}{1}, \frac{0}{2}, \frac{1}{2}, \frac{0}{3}, \frac{1}{3}, \frac{2}{3}, \dots , \frac{0}{k}, \frac{1}{k}, \dots , \frac{k-1}{k}$ is equidistributed in the ...
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On the convergence rate of $\sum \frac{\ln(n)}{n}\{x^n+x^{-n}\}$
Let
$$S:=\left\{x\in \mathbb{R}:\sum \frac{\ln(n)}{n} \{x^n+x^{-n} \}<+\infty\right\}\\
S':=S\cap(1,+\infty)$$
In this post it was proved that $\mu(S)=0$, thanks to the equidistribution of $\{x^n\}...
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Does equidistribution imply convergence
The following is an interesting problem presented to this site which has yet to bet solved: Does
$$\sum_{n=1}^\infty \frac{\sin(n!)}{n}$$
converge. While attempting this problem, I thought that ...
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$\sqrt2 n^2$ is uniformly distributed. [duplicate]
How to prove $(\sqrt2 n^2)_{n\in \mathbb{N}}$ is uniformly distributed mod 1.
I tried using Weyl's criterion but I seem to go nowhere.
Thank You very much.
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Strange property of irrational numbers, linked to continued fractions
Convergents of continued fractions provide in some sense the best rational approximations to an irrational number. In my research trying to prove that the binary digits of $\sqrt{2}$ are 50% one, 50% ...
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Potential enhancement of the equidistribution theorem
The sequence $\{h_n \alpha\}$ is equidistributed mod $1$ if $\alpha$ is irrational and $h_n = n$. Is there a generalization of the equidistribution theorem with some sequence $h_n$ (other than the ...
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If $\sqrt{h_{n+1}}-\sqrt{h_n} \rightarrow 0$ and $\alpha$ irrational, then $\{h_n \alpha\}$ is equidistributed mod 1
Here $h_n$ is a sequence of increasing integer numbers. The brackets represent the fractional part function. I am looking for a reference about this statement (which is obvious if $h_n = n$), or a ...
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Effective equidistribution inequality for $\sqrt{2}$
Let $A=\bigg\lbrace k \geq 1 \ \bigg| \ \lbrace k\sqrt{2} \rbrace \lt \frac{1}{2}\bigg\rbrace$ (where $\lbrace x \rbrace$ denotes the fractional part of $x$), $a_n=|A\cap [1,n]|$ and $d_n=\frac{a_n}{n}...
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Comparing two rational approximation of the same minimum
Let $P(x)=x^3-6x$. It is easy to see that on $[1,2]$, $P$ reaches its minimum at $x=\sqrt{2}$. Among all the fractions with denominator dividing $n$, the two closest ones are $a_n=\frac{\lfloor n\sqrt{...
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Fourier analysis an Introduction Chap 4. Exercise 5
Prove that the sequence {$\gamma_{n}$}$_{n=1}^{\infty},$ where $\gamma_{n}$ is the fractional part of
$$\bigl( \frac{1+\sqrt{5}}{2}\bigr)^{n},$$
is not equidistributed in $[0,1].$
[Hint: Show that $...
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Equidistribution summing over the euclidean ball
Given a vector $v\in \mathbb{Z}^d\setminus\{0\}$, an irrational number $\eta$ and some $M>0$ is it true that$$\Big(\frac{\sqrt{d}}{M}\Big)^{d}\sum_{w\in \mathbb{Z}^d\cap B(0, M)}\exp(2\pi i \eta \...
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equality of fractional parts
I am asking myself a question. Let $\alpha > 1$ and $\{x\}$ denote the fractional part of $x$ which is $x - \lfloor x \rfloor$. Let $\{ u_n(x) =\alpha^n x \}_{n \in \mathbb{N^{*}}}$ Given $x \in [-\...
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Detail in van der Corput inequality
I still stuck on a detail of a proof of van der corput inequality.
Let $\left(x_1 , x_2 , ..., x_N\right) \in \mathbb{C^{N}}$ and $1 \leq H \leq N$ an integer. Let us consider the sum $\Sigma_{2} = \...
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reduce to the case $[0,1]$ in equidistribution modulo $1$
I am trying to prove a theorem using an optimal solution. Given $(y_n)_{n \in \mathbb{N^{*}}}$ such that $\exists \delta > 0, \forall n \in \mathbb{N^{*}} y_{n+1} - y_n \geq \delta$ then for almost ...
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If $\{a_n\},\{b_n\}$ are equidistributed, is $\{a_n\}\cup\{b_n\}$ equidistributed?
If $\{a_n\},\{b_n\}$ are equidistributed, is $\{a_n\}\cup\{b_n\}$ equidistributed ?
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Equidistribution of $\{p_n^2 \alpha \}$
Let $p_n$ be the $n$th prime and $\alpha$ an irrational number. Vinogradov proved that the sequence $\{p_n \alpha \}$ is equidistributed. Is it known whether the sequence $\{p_n^2 \alpha \}$ is ...
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