Questions tagged [equidistribution]

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Equidistribution on $\mathrm{SU}_2$

Let $F_{a_1,a_2}$ be the free group with a free generating set $\{a_1,a_2\}$ of two elements, and for any $n\in\mathbb{N}$, set $A_n=\{\text{reduced words in } F_{a_1,a_2} \text{with length} \leqslant ...
Local's user avatar
  • 86
4 votes
2 answers
423 views

Reference request - Pillai-Selberg Theorem

I want to find a proof of the following claim. Let $\Omega(n)$ denote the number of prime factors of an integer $n$ counted with multiplicity. Then $\Omega(n)$ equidistributes over residue classes. ...
alidixon222's user avatar
4 votes
0 answers
120 views

Vanishing exponential sums of fractional parts of polynomials

Let $p$ be an integer polynomial and $k$ be a natural number, both fixed. Is it the case that if $$C(\alpha) = \sum_{i=1}^k e(\alpha \{p(i)/k\})$$ equals 0, then $\alpha$ is an integer? Here, $e(x) = ...
Borys Kuca's user avatar
1 vote
1 answer
84 views

Is equal natural density on intervals with matching areas but opposite signs sufficient to use fixed-width part sizes for a simple Riemann sum?

Suppose we have a sequence $\theta_n$ which is dense on $\left(0,2\pi\right)$. Furthermore, if $A=(x,y)\subset(0,\pi)$ and $B=(x+\pi,y+\pi)$ for some $x,y$, and if we define the natural density of a ...
Sky Waterpeace's user avatar
14 votes
2 answers
542 views

for which $A\subset \mathbb{N}$ does there exist irrational $\alpha$ for which $\alpha\cdot A \pmod 1$ is not uniformly distributed?

In this question it is discussed that for a set $$A=\{2^n3^m| n,m=0,1,2,\ldots\}$$ there exists an irrational number $\alpha$ such that the set $\alpha\cdot A$ (being naturally enumerated) is not ...
Fedor Petrov's user avatar
7 votes
2 answers
382 views

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 $\alpha$ such ...
Miranda's user avatar
  • 173
3 votes
0 answers
109 views

How do you ensure that the product of two power series is equidistributed?

Let $f(T) = \sum a_n T^n \in \mathbf{F}_p [[ T ]]$ be a power series. We'll say that $f(T)$ is equidistributed if for every $a \in \mathbf{F}_p$, we have $$\lim_{X \to \infty} \dfrac{1}{X} \cdot \# \{...
Adithya Chakravarthy's user avatar
8 votes
1 answer
586 views

Is the product of two equidistributed power series equidistributed?

Let $f(T) = \sum a_n T^n \in \mathbf{F}_p [[ T ]]$ be a power series. We'll say that the coefficients of $f(T)$ are equidistributed modulo $p$ if for every residue class $a$ modulo $p$, we have $$ \...
Adithya Chakravarthy's user avatar
10 votes
1 answer
705 views

Continuous variant of the Chinese remainder theorem

Let $x_1,x_2,\ldots, x_k \in [0,1]$ be irrational numbers. I'm interested in what, if anything, can be said about the values $\{nx_i \bmod 1: n\in \mathbb{N}\}$. Specifically, I'm interested if there ...
Alek Westover's user avatar
10 votes
2 answers
352 views

Behavior of $\sum_{n=1}^{\infty} (\{n \xi \} - \frac{1}{2})$ for irrational number $\xi$

Consider the series $$ \sum_{n=1}^{\infty} ( \{ n \xi \} - \frac{1}{2}) $$ where $\{ \}$ denotes the fractional part and $\xi$ is some irrational number. It clearly does not converge but can we show ...
Hampus Nyberg's user avatar
2 votes
0 answers
124 views

Distribution of square roots (mod m) on small intervals (with respect to m)

Fix a large positive integer $m$. Let $A$ be small positive number typically $\sim m^{1/3}$. Suppose $S(A, m)$ be set of solutions (normalized by dividing by $m$) to the quadratic congruences $x^2 = a ...
Melanka's user avatar
  • 577
10 votes
1 answer
716 views

Cancellation in a very rapidly oscillating exponential sum

Let $f(x)$ be a function such that for all $T \leq f(x), \ T / x \to \infty$ we have $$ \sum_{x \leq n < 2 x} e^{2 \pi i T / n} = o(x). $$ How fast can $f(x)$ grow? I can show that for any $\...
Random's user avatar
  • 2,374
0 votes
0 answers
49 views

Product-based binary numeration system

I am looking at the following binary numeration system: $$x =\prod_{k=1}^\infty \Bigg(1+\frac{d_k(x))}{2^k}\Bigg), \quad d_k(x)\in \{0, 1\}.$$ The $d_k$'s are the digits, and $x$ is between $1$ (all ...
Vincent Granville's user avatar
7 votes
2 answers
784 views

Convolution sum of divisor functions

Let $\sigma_0(n)$ be the divisor counting function $$\sigma_0(n) = \sum_{d \vert n} 1.$$ I'm interested in the convolution sum $$ S(n) := \sum_{k=1}^{n-1} \sigma_0(k) \sigma_0(n-k)$$ I ran some quick ...
Adithya Chakravarthy's user avatar
7 votes
3 answers
578 views

Is the sequence $\{n/\{ \sqrt{2}n\} \}$ uniformly distributed in $[0,1)$?

A classical theorem of Kronecker says that the sequence $(\{\alpha_1 n\}, \{\alpha_2 n\},\dots,\{\alpha_d n\})$ ($n \in \mathbb{N}$) is uniformly distributed in $[0,1)^d$ provided that $1,\alpha_1,\...
Jakub Konieczny's user avatar
9 votes
1 answer
500 views

Is the divisor counting function equidistributed mod $p$?

Let $\sigma_0(n)$ be the divisor counting function: $$\sigma_0(n) = \sum_{d \vert n} 1.$$ I ran some numerical experiments that showed when $p$ is prime, the function $\sigma_0(n)$ is equidistributed ...
Adithya Chakravarthy's user avatar
0 votes
1 answer
136 views

Equidistribution and moments

Equidistribution of a set sequence 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 ...
nervxxx's user avatar
  • 207
2 votes
0 answers
53 views

Equidistribution of lattice points on quadratic forms without certain values

I have recently been studying some results about equidistribution of lattice points on positive definite quadratic forms $Q$ with $3$ or more variables. Concretely the article Duke and Schulze-Pillot, ...
MathqA's user avatar
  • 313
4 votes
1 answer
264 views

Equidistribution of distances of integer points to a circle

I have noticed in the following graph that the euclidean distance between points $k \in\mathbb{Z}^2\cap C_7^1$ ($C_7^1:={}$Circle with radius 7 and shell with thickness 1) and the nearest point on the ...
HyyFly's user avatar
  • 187
3 votes
1 answer
206 views

Duke and Schulze-Pillot condition for equidistribution

When regarding a ternary quadratic form $Q(x,y,z)$, is a classic question to consider which integers $n$ can be represented by $Q$. It is also classic to wonder how "well distributed" are ...
MathqA's user avatar
  • 313
4 votes
1 answer
344 views

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}{\...
Joshua Stucky's user avatar
1 vote
0 answers
166 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 ...
asrxiiviii's user avatar
1 vote
1 answer
125 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 ...
Melanka's user avatar
  • 577
8 votes
2 answers
391 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 ...
H A Helfgott's user avatar
  • 19.3k
5 votes
0 answers
208 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 ...
Adithya Chakravarthy's user avatar
1 vote
1 answer
264 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'...
Cloudscape's user avatar
3 votes
0 answers
150 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 ...
Farzad Aryan's user avatar
3 votes
1 answer
651 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=...
Vincent Granville's user avatar
3 votes
0 answers
50 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 ...
Roland Bacher's user avatar
1 vote
2 answers
163 views

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)...
Arsh Jhaj's user avatar
  • 123
10 votes
1 answer
570 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 ...
Erik4's user avatar
  • 121
2 votes
0 answers
195 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 ...
Christopher D. Long's user avatar
2 votes
0 answers
399 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 ...
Kavan Prajapati's user avatar
2 votes
1 answer
471 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 ...
Nate River's user avatar
  • 4,802
13 votes
1 answer
2k 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}...
Nate River's user avatar
  • 4,802
1 vote
0 answers
85 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+...
katago's user avatar
  • 543
0 votes
2 answers
147 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 ...
Tartrate's user avatar
  • 341
5 votes
2 answers
549 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 $\{n\alpha\}$, $\{n^2\alpha\}$ ...
Vincent Granville's user avatar
4 votes
0 answers
127 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 $\...
GWB's user avatar
  • 181
12 votes
1 answer
485 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 ...
Daniel Loughran's user avatar
1 vote
1 answer
108 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 ...
Kevin Smith's user avatar
  • 2,470
2 votes
1 answer
146 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) ={ ...
Ivan Meir's user avatar
  • 4,762
2 votes
1 answer
382 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 ...
Schmelli's user avatar
1 vote
0 answers
127 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 $\...
Anatoly's user avatar
  • 163
0 votes
1 answer
127 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 ...
user142929's user avatar
1 vote
0 answers
67 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 \...
user86558's user avatar
7 votes
1 answer
631 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 ...
Jean's user avatar
  • 515
15 votes
1 answer
1k 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 ...
José Navarro's user avatar
6 votes
1 answer
451 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 ...
zoidberg's user avatar
  • 200
3 votes
0 answers
151 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,...
Turbo's user avatar
  • 13.6k