# 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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### 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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### 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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### 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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### 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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