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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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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Show that $\lim_{N \to\infty }\int_{0}^{1}\left | \frac{\sum_{n=1}^{N}f(x+\xi _{n})}{N} \right |^{2}dx=0$ in which $(\xi _{n})$ is equidistributed

Suppose that f is a periodic function on $R$ of period $1$, and $(\xi _{n})$ is a sequence which is equidistributed in $[0,1)$, and assume that f is riemann integrable and $\int_{0}^{1}f(x)dx=0$,then ...
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How to prove the following statement of an equidistributed sequence?

Suppose that $\xi _{n}$ is an equidistributed sequence, f is a periodic integrable function of period $1$ on $R$, and $\int_{0}^{1}fdx=0$, then $\lim_{N\rightarrow \infty }\int_{0}^{1}\left | \frac{\...
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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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Katz-Sarnak results on equidistribution of coefficients of Frobenius of curves over finite fields

I wonder if there is some way to prove the following result: given a (possibly big) genus $g$ and a prime number $p$, let $k$ be the field with $p$ elements. Let $S_p$ be the set of all the possibile ...
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Is it possible to have a closed formula for a sequence 2, 4, 6, 8, ... and the sequence would be equidistributed?

I've seen on this site questions asking about rules which would generate a sequence which deviates from say, $2n$ and generate different sequence up to infinity. Now, I know this is possible. What ...
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Generic point - invariant measure

A point $x \in X$ is $\mu$ - generic if $$\lim \frac{1}{n} \sum_{j=0}^{n-1} \delta_{T^{n}x} = \mu$$ (limit in the weak sense). Take, for example, the following construction: $X = \{0,1\}^{\mathbb{Z}}...
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4 votes
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Is the sequence equidistributed on the sphere?

Let $\Gamma=\{\vec{h}\in \mathbb{Z}^3: \vec{h}\neq \vec{0}\}$ and let $\{ \vec{h}_n\}_{n\geq 1}$ be an ordering of the elements of $\Gamma$ such that $\Gamma=\{\vec{h}_n : n\geq 1\}$ and for all $n, m$...
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Equi-distributed sequences for a doubling map?

Let us consider the doubling map $\tau:X:=[0,1)\to X:=[0,1)$ defined by $$\tau(x):= 2x-\lfloor 2x \rfloor=2x\,\text{mod}\,1 $$ It is well-known that $\tau$ is ergodic and hence for any continuous ...
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Equidistributed sequence in all $\Bbb R $?

Does there exist a real sequence $(x_n)$ such that $$\forall \ a < b,\ c < d \in \Bbb R$$ $$ \lim_{n\to \infty}\frac{\operatorname{Card}\{0 ≤ i ≤n\ :\ a≤x_i≤b\}}{\operatorname{Card}\{0 ≤ i ≤n\ ...
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12 votes
2 answers
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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 ...
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8 votes
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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 $x^2-2y^2=1(x,y\in\mathbb N)$ ...
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Confusion about line in proof that (log p(n)) is not equidistributed

The following question is about a well-known note posted here. The note is nicely written and I am hoping that clarification of one line on page 35 will make things fall into place. The line is: $$\...
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A property related to equidistribution of sequences

One version of the equidistribution property raised 2 questions that I am trying to answer. Let a sequence $\{s_1,s_2,s_3,... \}$be equidistributed (ED) on an interval $[a,b]$ if for any subinterval $[...
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