Questions tagged [equidistribution]
The equidistribution tag has no usage guidance.
61
questions
3
votes
2
answers
249
views
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 ...
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. ...
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) = ...
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 ...
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 ...
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 ...
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
$$\lim_{X \to \infty} \dfrac{1}{X} \cdot \# \{...
8
votes
1
answer
586
views
Is the product of two equidistributed power series equidistributed?
Let $f(T) = \sum a_n T^n \in \mathbf{F}_p [[ T ]]$ be a power series. We'll say that the coefficients of $f(T)$ are equidistributed modulo $p$ if for every residue class $a$ modulo $p$, we have
$$ \...
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 ...
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 ...
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 ...
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 $\...
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 ...
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 ...
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,\...
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 ...
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 ...
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, ...
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 ...
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 ...
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
$$
\tag{1}
\pi(x;q,a) \leq \frac{2x}{\...
1
vote
0
answers
166
views
Uniform distribution mod $1$ vs independence of random variables
Let $a_1, \cdots, a_k \in [0, 1)$ be real numbers such that $1, a_1, \cdots, a_k$ are independent over the rational numbers. By the Weyl equidistribution criterion in $k$-dimensions, we know that the ...
1
vote
1
answer
125
views
Distribution of quadratic polynomials mod $n$ and $n^2$
Suppose $n$ is odd, then both equations $x^2 = D \; mod \;n$ and $x^2 = D \; mod \;n^2$ have the same number of solutions for fixed $D$ coprime to $n$. What can be said about the relationship between ...
8
votes
2
answers
391
views
Distribution of $\{x/n^2\}$
Let $x$ be a large positive real number. Let $I$ be an interval -- say, $I=[1,\sqrt{\epsilon x}]$. Let $n$ range over the integers in $I$, or over the intersection of $I$ and an arithmetic progression ...
5
votes
0
answers
208
views
Are the $p$-adic digits of roots of unity equidistributed?
I was looking at the $p$-adic expansions of roots of unity in $\mathbf{Z}_p$, and I noticed that the digits tended to be equidistributed among the numbers $\{0, 1, \dots, p-1 \}$. I wanted to ask if ...
1
vote
1
answer
264
views
The lonely runner conjecture and equidistribution on tori
So, I've been reading about the lonely runner conjecture, but I have to admit that my knowledge of diophantine approximation is so limited that I wouldn't be able to properly explain the term.
So let'...
3
votes
0
answers
150
views
Erdős–Turán inequality for complex numbers
Consider the following set of complex numbers in the upper half plane: $$\{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\}.$$
Assume that this set ...
3
votes
1
answer
651
views
$\{(\log n)^\alpha\}$ not equidistributed if $0<\alpha\leq 1$, so how is it distributed?
The brackets denote the fractional part function. It is well known that the distribution (defined as the limit of the empirical distribution) is $F(x)=(e^x - 1)/(e-1)$, with $x\in [0, 1]$, if $\alpha=...
3
votes
0
answers
50
views
Asymptotical equidistribution of index $p$ subgroups of $\mathbb Z^2$ on the unit tangent bundle of the modular curve
Given a prime $p$ we get $p+1$ sublattices of index $p$ in $\mathbb Z^2$ (identified with $\mathbb Z[i]\subset \mathbb C$) which correspond to some points on the moduli space of such
lattices up to ...
1
vote
2
answers
163
views
Estimates on the discrepancy of random sequences
The discrepancy of a $[0,1]$-valued sequence $n \mapsto \alpha_n$ is the quantity $$D(N; \alpha) \stackrel{\text{def}}{=} \sup_{(a,b) \subset [0,1]} \left|\frac{\#\{1 \leq n \leq N : \alpha_n \in (a,b)...
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 ...
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 ...
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 ...
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 ...
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}...
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+...
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 ...
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 $\{n\alpha\}$, $\{n^2\alpha\}$ ...
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 $\...
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 ...
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 ...
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 $I\subset [0,1]$,
$$E(n,\theta, I) ={ ...
2
votes
1
answer
382
views
Is this a criterion for uniform distribution modulo one?
For all $k \in \mathbb N$ let $a_k$ be strictly positive bounded weights, i.e. there are constants $C_1$ and $C_2$ such that $0<C_1\le a_k \le C_2$. Now a real valued sequence $(x_k)_{k \in \mathbb ...
1
vote
0
answers
127
views
Convergence rate of equidistributed sequences
In the sum
$$\sum_{n=1}^N \left(\{n \alpha\}-\frac{1}{2}\right)$$
where $\{x\}$ indicates the fractional part of $x$ and $\alpha$ is an irrational number, Koksma inequality suggests an order of $\...
0
votes
1
answer
127
views
The series $\sum_{n=1}^\infty {2n\brace n}^{-{2n\brace n}}$ and $\sum_{n=1}^\infty (2n)_{n}^{-(2n)_{n}}$ in the context of normal numbers
In this ocassion we consider the followgin series that involve ${n\brace k}$ the Stirling number of the second kind and $(n)_k$ the Pochhammer symbols. I've known from an informative point of view ...
1
vote
0
answers
67
views
Equidistribution of linear forms over euclidean ball
Given a vector $v\in \mathbb{Z}^d\setminus\{0\}$, an irrational number $\eta$ and some big $M>0$ what type of bound can one get on $$\sum_{w\in \mathbb{Z}^d\cap B(0, M)}\exp(2\pi i \eta \cdot \...
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 ...
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 ...
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 ...
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,...