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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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Does half of the sequence $\left(n^{\alpha}\right)_{n=1}^{\infty}$ have even integer part/floor?

Define $e\left((a_n)_{n=1}^{\infty};N\right)$ to be the amount of members of the sequence $(a_n)_{n=1}^{\infty}$ that are $\leq N$ and have even integer part (also known as the floor of the number). ...
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Approximation of $1/2$ by fractional parts

Let $\{u\}$ denote the fractional part of a real $u \geq 0$, and $\mathbb{N} = \{0, 1, 2, \dots\}$. For positive integer $N$, define $$ T_N(x) = \sup_{k \in \mathbb{N}} \min_{m \in [kN, (k+1)N) \cap \...
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Is $\{n\log n \pmod 1: n\in\mathbb{N} \}$ dense in $[0,1]?$ If so, is it uniformly distributed?

It is clear that $\{\log n\pmod 1: n\in\mathbb{N} \}$ is dense in $[0,1]$ but not uniformly distributed. How about $\{n\log n \pmod 1: n\in\mathbb{N} \} ?$ Is it dense in $[0,1]?$ If so, is it ...
Adam Rubinson's user avatar
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For each $n\in\mathbb{N},$ let $x_n:=\min_{1\leq k < n}\lvert\sin n-\sin k\rvert.\ $ Does $\sum_{n=1}^{\infty} x_n $ converge?

For each $n\in\mathbb{N},$ let $x_n:= \displaystyle\min_{1\leq k < n} \lvert\sin n - \sin k\rvert.\ $ Does $\displaystyle\sum_{n=1}^{\infty} x_n $ converge? Consider instead, $a_1 = 0,\ a_2=1, ...
Adam Rubinson's user avatar
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Stuck on proof equidistribution on homogeneous space

I have been trying to understand the following proof but I don't fully understand the implicit last steps. From the replacement of the groups to the deducing of the theorems. I wondered if anyone ...
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Intuition behind Weyl's equidistribution theorem

Recently I've been studying Fourier analysis through Stein's book, and there is a section there dedicated to Weyl's Equidistribution Theorem, specifically, A sequence $(\xi_n)_n$ in $[0, 1)$ is ...
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Questions regarding the function $\Omega(n)$

For any positive integer $n$, define $\Omega(n)$ to be the number of prime factors (including repeated factors, so for example $\Omega(12)=\Omega(2^2\times 3)=3$). It is well known (Pillai-Selberg) ...
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Question regarding behaviour of equidistributed sequences.

If $(s_1, s_2, s_3\ldots )$ is an equidistributed sequence on $[0,1],$ then for each $\ 0<\delta<\varepsilon <1\ $ and each $\ c\in [0,1-\varepsilon],\ \exists\ N\ $ such that $$ \varepsilon -...
Adam Rubinson's user avatar
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$(\Delta^{(\infty)})^{p+1} \leq N(p+1)\Delta^{(p)}$ i.e. Equivalence of $p$-discrepancies

Let $\xi_1,\xi_2,\dots,\xi_N \in \mathbb{R}$, $p \geq 1$, I want to show $(\Delta^{(\infty)}(\xi_1,\dots,\xi_N))^{p+1} \leq N(p+1)\Delta^{(p)}(\xi,\dots,\xi_N)$ Definitions : Let $\psi(x) := \begin{...
Paul's user avatar
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Let $(a_{n})_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the set$\{2^n3^m\mid m,n\in N\}$. Find α such that $(a_n)\alpha\pmod1$ is not equidistributed.

Let $$(a_{n})_{n \in \mathbb{N}} = (1,2,3,4,6,8,9,12,16,18,\cdots)$$ be a sequence that is a listing of the set $$\{2^n3^m \mid m,n \in \mathbb{N}\}$$ We need to find an irrational number $\alpha$ ...
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General proposition that, if true, would immediately prove that $\sum\sin n$ and $\sum\cos n$ are bounded

I know that the sequence $A_n = \displaystyle\sum_{k=1}^n \sin k$ is bounded. I suspect the following general proposition is true, and it could then immediately imply $A_n$ is bounded, as an ...
Adam Rubinson's user avatar
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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 $...
Adithya Chakravarthy's user avatar
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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 ...
Adam Rubinson's user avatar
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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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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 ...
Sayan Dutta's user avatar
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4 votes
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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 ...
Adam Rubinson's user avatar
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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 $n$ ...
Matt F.'s user avatar
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2 answers
252 views

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 $S(x)=\sum_\limits{n=1}^x\left(|\sin(n)|-\frac{2}{\pi}\right)$....
te4's user avatar
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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}^{...
Petra Axolotl's user avatar
2 votes
1 answer
306 views

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 \...
Adam Rubinson's user avatar
2 votes
0 answers
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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 ...
Killua Zoldyck's user avatar
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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 :$...
Killua Zoldyck's user avatar
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1 answer
111 views

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 $|...
amsmath's user avatar
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2 votes
1 answer
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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}^...
HooMun's user avatar
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1 answer
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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, $\{ \langle k\...
wzstrong's user avatar
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1 answer
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${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 $α\in ...
Rick Sanchez C-666's user avatar
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Condition for ({$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 ...
x100c's user avatar
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1 vote
1 answer
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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 $g \in \Bbb F_p^{\times}$ (...
Watson's user avatar
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2 votes
1 answer
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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 $1$ ...
Anvit's user avatar
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7 votes
1 answer
495 views

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 ...
richrow's user avatar
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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 ...
Canjioh's user avatar
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0 answers
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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 ...
J.Mayol's user avatar
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2 votes
1 answer
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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\}...
EBloch's user avatar
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7 votes
1 answer
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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 ...
QC_QAOA's user avatar
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2 votes
1 answer
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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.
epsilon_delta's user avatar
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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% ...
Vincent Granville's user avatar
1 vote
1 answer
78 views

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 ...
Vincent Granville's user avatar