# 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, ...
1 vote
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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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1 vote
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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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### 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.
1 vote
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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% ...
1 vote
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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 ...
1 vote