Questions tagged [equidistribution]

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Generalizations of the Brun-Titchmarsh theorem

Let $\pi(x;q,a)$ count the number of primes $\leq x$ congruent to $a$ mod $q$. The Brun-Titchmarsh Theorem states that for all $q< x$, $(a,q)=1$, we have $$ \tag{1} \pi(x;q,a) \leq \frac{2x}{\...
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1 vote
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72 views

Uniform distribution mod $1$ vs independence of random variables

Let $a_1, \cdots, a_k \in [0, 1)$ be real numbers such that $1, a_1, \cdots, a_k$ are independent over the rational numbers. By the Weyl equidistribution criterion in $k$-dimensions, we know that the ...
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1 vote
1 answer
105 views

Distribution of quadratic polynomials mod $n$ and $n^2$

Suppose $n$ is odd, then both equations $x^2 = D \; mod \;n$ and $x^2 = D \; mod \;n^2$ have the same number of solutions for fixed $D$ coprime to $n$. What can be said about the relationship between ...
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8 votes
2 answers
360 views

Distribution of $\{x/n^2\}$

Let $x$ be a large positive real number. Let $I$ be an interval -- say, $I=[1,\sqrt{\epsilon x}]$. Let $n$ range over the integers in $I$, or over the intersection of $I$ and an arithmetic progression ...
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5 votes
0 answers
130 views

Are the $p$-adic digits of roots of unity equidistributed?

I was looking at the $p$-adic expansions of roots of unity in $\mathbf{Z}_p$, and I noticed that the digits tended to be equidistributed among the numbers $\{0, 1, \dots, p-1 \}$. I wanted to ask if ...
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  • 695
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1 answer
190 views

The lonely runner conjecture and equidistribution on tori

So, I've been reading about the lonely runner conjecture, but I have to admit that my knowledge of diophantine approximation is so limited that I wouldn't be able to properly explain the term. So let'...
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3 votes
0 answers
107 views

Erdős–Turán inequality for complex numbers

Consider the following set of complex numbers in the upper half plane: $$\{ic_n \pm \gamma_n: 0 \leq n \leq N, \hspace{1 mm} c_0=\gamma_0=0, c_n,\hspace{1 mm} \gamma_n>0\}.$$ Assume that this set ...
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2 votes
1 answer
484 views

$\{(\log n)^\alpha\}$ not equidistributed if $0<\alpha\leq 1$, so how is it distributed?

The brackets denote the fractional part function. It is well known that the distribution (defined as the limit of the empirical distribution) is $F(x)=(e^x - 1)/(e-1)$, with $x\in [0, 1]$, if $\alpha=...
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3 votes
0 answers
47 views

Asymptotical equidistribution of index $p$ subgroups of $\mathbb Z^2$ on the unit tangent bundle of the modular curve

Given a prime $p$ we get $p+1$ sublattices of index $p$ in $\mathbb Z^2$ (identified with $\mathbb Z[i]\subset \mathbb C$) which correspond to some points on the moduli space of such lattices up to ...
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1 vote
1 answer
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Estimates on the discrepancy of random sequences

The discrepancy of a $[0,1]$-valued sequence $n \mapsto \alpha_n$ is the quantity $$D(N; \alpha) \stackrel{\text{def}}{=} \sup_{(a,b) \subset [0,1]} \left|\frac{\#\{1 \leq n \leq N : \alpha_n \in (a,b)...
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314 views

Upper bound for maximum modulus of polynomial on unit circle in term of the distribution of its roots

Let $P(z) = \prod_{i = 1}^n (z - z_i) \in \mathbb{C}[z]$ be a monic polynomial having all roots $z_1, \dots, z_n$ on the unit circle $\mathbb{T} := \{z \in \mathbb{C} : |z| = 1\}$. What is known about ...
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  • 111
2 votes
0 answers
113 views

Uniform distribution of log(log((n!)!)) mod 1

Can it be shown that the sequence $\log(\log((n!)!))$ is uniformly distributed mod 1? I believe it should be, but I'm not certain if Stirling's approximation repeated twice combined with the ...
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2 votes
0 answers
301 views

Conjecture: The sequence {$π(2n+1)!$} is equidistributed in the interval (0,1)

Let $n\in\mathbb{N}$. From the book "Uniform Distribution of Sequences" (available here) by L. Kuipers and H. Niederreiter, (from pg. 8) I found that for any irrational $\theta$, the ...
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2 votes
1 answer
217 views

Writing the Lebesgue–Stieltjes integral as a sum of equidistributed Dirac delta measures

Problem set up: Let $f: [0, 1] \to \mathbb R$ be an absolutely continuous function (thus a fortiori of bounded variation) such that its total variation on any open interval $(a, b)$ is $b-a$. We say a ...
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12 votes
1 answer
1k views

Anti Arzela-Ascoli

Notation: We say a sequence of real numbers diverges if it does not converge to a finite limit. We say a sequence $f_n$ of real valued functions on $[0, 1] $ are equibounded if $\sup_{n \in \mathbb N}...
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1 vote
0 answers
76 views

How fast will it converge to equilibrium?

$\alpha \in \mathbb{R} \backslash \mathbb{Q}$, given $\varepsilon>0 \quad \lambda>0$. given $n_{1}, \cdots, n_{k} \in N^{*}$ satisfied $(1-\varepsilon) \lambda<n_{1}<\cdots<n_{k}<(1+...
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  • 453
0 votes
2 answers
86 views

Equidistributed sequence wrt exponential/Gaussian measure

For an arbitrary probability space $(X,\mu)$, a sequence $(x_n)$ in $X$ is said to be equidistributed with respect to $\mu$ if the measures $\frac 1 n \sum_{1\le k\le n} \delta_{x_k}$ converges weakly ...
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  • 321
5 votes
2 answers
437 views

Sequences similar to $\{n\alpha\}$ that are both equidistributed and truly random-like

See update at the bottom. Here the brackets represent the fractional part, and $\alpha \in [0, 1]$ is a positive irrational number. It is well known that the sequences $\{n\alpha\}$, $\{n^2\alpha\}$ ...
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4 votes
0 answers
109 views

$\delta$-equidistributed polynomials over finite fields

I'm trying to show that a polynomial over finite (prime) field is "close enough" to being equidistributed over its range. A polynomial $p(\cdot)$ from $\mathbb{F}^n$ to $\mathbb{F}$ is $\...
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  • 161
12 votes
1 answer
400 views

Equidistribution of $\{\alpha p\}$ for $p$ in an arithmetic progression

Let $\alpha$ be irrational. A famous theorem of Vinogradov says that $\{ \alpha p\}$ is equidistributed in $[0,1]$ as $p$ runs over all primes. Let $a,q$ be natural numbers with $\gcd(a,q) = 1$. Then ...
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1 vote
1 answer
91 views

How do solutions to this quadratic congruence distribute as the number of factors grows?

This is a question that arose during a conversation with a colleague regarding Landau's fourth problem, which asks whether there are infinitely many primes of the form $n^2+1$. The Conjectured ...
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  • 2,440
2 votes
1 answer
101 views

O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval

O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval. Using $\{x\}$ to denote the fraction part of $x$ we can define for any $I\subset [0,1]$, $$E(n,\theta, I) ={ ...
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  • 4,350
2 votes
1 answer
368 views

Is this a criterion for uniform distribution modulo one?

For all $k \in \mathbb N$ let $a_k$ be strictly positive bounded weights, i.e. there are constants $C_1$ and $C_2$ such that $0<C_1\le a_k \le C_2$. Now a real valued sequence $(x_k)_{k \in \mathbb ...
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1 vote
0 answers
95 views

Convergence rate of equidistributed sequences

In the sum $$\sum_{n=1}^N \left(\{n \alpha\}-\frac{1}{2}\right)$$ where $\{x\}$ indicates the fractional part of $x$ and $\alpha$ is an irrational number, Koksma inequality suggests an order of $\...
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  • 163
0 votes
1 answer
102 views

The series $\sum_{n=1}^\infty {2n\brace n}^{-{2n\brace n}}$ and $\sum_{n=1}^\infty (2n)_{n}^{-(2n)_{n}}$ in the context of normal numbers

In this ocassion we consider the followgin series that involve ${n\brace k}$ the Stirling number of the second kind and $(n)_k$ the Pochhammer symbols. I've known from an informative point of view ...
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1 vote
0 answers
59 views

Equidistribution of linear forms over euclidean ball

Given a vector $v\in \mathbb{Z}^d\setminus\{0\}$, an irrational number $\eta$ and some big $M>0$ what type of bound can one get on $$\sum_{w\in \mathbb{Z}^d\cap B(0, M)}\exp(2\pi i \eta \cdot \...
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7 votes
1 answer
592 views

Digits in an algebraic irrational number

I am trying to solve a problem and I got a conditional result related to normality of algebraic irrational numbers (Borel conjecture). I know that by using Ridout theorem or Schmidt subspace theorem ...
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  • 355
14 votes
1 answer
974 views

Uniform distribution of points on Riemannian manifolds

Recently, I came across a beautiful paper by Arnol'd and Krylov (Uniform distribution of points on a sphere...) that contains the following theorem: Theorem: Let A and B be two rotations of the ...
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6 votes
1 answer
335 views

Equidistribution of $\{p_n^2α\}$

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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  • 200
3 votes
0 answers
146 views

Probability distribution from equidistribution - I

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ norm of vector $(x,y)$ as $(x,y)$ ranges over all non-zero integral solutions to $(x,y)\equiv t(a,...
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  • 12.8k
14 votes
2 answers
494 views

Equidistribution of CM points in the principal genus

It is well known that as the negative discriminant $-D$ goes to infinity, the number of quadratic forms of discriminant $-D$ belonging to the principal genus also goes to infinity. Can we say ...
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4 votes
1 answer
349 views

Explicit numbers with square root cancellation in Weyl's exponential sum

I'm interested in examples of real numbers $\alpha$ where we have $$\left| \sum_{n=1}^N \mathrm e(\alpha n) \right| \ll N^{1/2} $$ or perhaps with the weaker estimate with the right side replaced ...
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53 votes
5 answers
3k views

Distribution of square roots mod 1

I was wondering about the distribution of $\sqrt{p}$ mod $1$ this morning, as one does while brushing one's teeth. I remembered the paper of Elkies and McMullen (Duke Math. J. 123 (2004), no. 1, 95–...
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1 vote
2 answers
157 views

A distribution related to Fermat's two squares theorem

Fermat's two squares theorem tell us that every prime number $p \equiv 1 \pmod 4$ can be written in a unique way as $p = a^2 + b^2$ for two positive integers $a < b$. In particular, we can ...
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10 votes
2 answers
310 views

Refined equidistribution for the periodic trajectories of Anosov flows?

Duke, and Linnik before him under a restrictive condition, proved that the set of closed geodesics of a given length $L$ is equidistributed on the modular surface as $L \to \infty$. This is a theorem ...
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3 votes
1 answer
70 views

Criterion for convergence of sums for non-continuous functions

The following question came up when thinking about equidistribution of Satake parameters of elliptic curves. Let $G$ be a compact Lie group with Haar measure $\mathrm{d} x$. Recall that a sequence $\{...
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8 votes
1 answer
400 views

Primes in arithmetic progression with a moduli equal to a power of 2

I am currently looking for a result stronger than Siegel-Walfisz theorem, which gives an upper bound on the error term $|\pi(x,a,b)-\frac{\pi(x)}{\phi(a)}|$ for particular $a$. The Siegel Walfisz is ...
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8 votes
1 answer
288 views

Angular distribution of zero sets of sparse polynomials

Consider a sequence of complex polynomials $f \in \mathbb{C}[z]$, $f(0) \neq 0$, that are composed of a negligible fraction $o(\deg{f})$ of monomials. Are the zeros of such polynomials necessarily ...
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6 votes
3 answers
834 views

Uniformly distributed sequence in $\mathbb{R}$

We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and $$\lim_{N \to \infty} \...
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  • 61
15 votes
5 answers
7k views

Elementary proof of the equidistribution theorem

I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is ...
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