Questions tagged [equidistribution]
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56
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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 ...
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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 ...
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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
$$\lim_{X \to \infty} \dfrac{1}{X} \cdot \# \{...
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1
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Is the product of two equidistributed power series equidistributed?
Let $f(T) = \sum a_n T^n \in \mathbf{F}_p [[ T ]]$ be a power series. We'll say that the coefficients of $f(T)$ are equidistributed modulo $p$ if for every residue class $a$ modulo $p$, we have
$$ \...
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votes
1
answer
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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 ...
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2
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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 ...
- 177
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votes
0
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119
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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 ...
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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 $\...
- 2,374
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votes
0
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47
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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 ...
- 3,058
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votes
2
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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 ...
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3
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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,\...
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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 ...
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1
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126
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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 ...
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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, ...
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1
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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 ...
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1
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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 ...
- 303
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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
$$
\tag{1}
\pi(x;q,a) \leq \frac{2x}{\...
- 1,668
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0
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Uniform distribution mod $1$ vs independence of random variables
Let $a_1, \cdots, a_k \in [0, 1)$ be real numbers such that $1, a_1, \cdots, a_k$ are independent over the rational numbers. By the Weyl equidistribution criterion in $k$-dimensions, we know that the ...
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Distribution of quadratic polynomials mod $n$ and $n^2$
Suppose $n$ is odd, then both equations $x^2 = D \; mod \;n$ and $x^2 = D \; mod \;n^2$ have the same number of solutions for fixed $D$ coprime to $n$. What can be said about the relationship between ...
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Distribution of $\{x/n^2\}$
Let $x$ be a large positive real number. Let $I$ be an interval -- say, $I=[1,\sqrt{\epsilon x}]$. Let $n$ range over the integers in $I$, or over the intersection of $I$ and an arithmetic progression ...
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Are the $p$-adic digits of roots of unity equidistributed?
I was looking at the $p$-adic expansions of roots of unity in $\mathbf{Z}_p$, and I noticed that the digits tended to be equidistributed among the numbers $\{0, 1, \dots, p-1 \}$. I wanted to ask if ...
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1
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The lonely runner conjecture and equidistribution on tori
So, I've been reading about the lonely runner conjecture, but I have to admit that my knowledge of diophantine approximation is so limited that I wouldn't be able to properly explain the term.
So let'...
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Erdős–Turán inequality for complex numbers
Consider the following set of complex numbers in the upper half plane: $$\{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\}.$$
Assume that this set ...
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$\{(\log n)^\alpha\}$ not equidistributed if $0<\alpha\leq 1$, so how is it distributed?
The brackets denote the fractional part function. It is well known that the distribution (defined as the limit of the empirical distribution) is $F(x)=(e^x - 1)/(e-1)$, with $x\in [0, 1]$, if $\alpha=...
- 3,058
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49
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Asymptotical equidistribution of index $p$ subgroups of $\mathbb Z^2$ on the unit tangent bundle of the modular curve
Given a prime $p$ we get $p+1$ sublattices of index $p$ in $\mathbb Z^2$ (identified with $\mathbb Z[i]\subset \mathbb C$) which correspond to some points on the moduli space of such
lattices up to ...
- 17k
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1
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Estimates on the discrepancy of random sequences
The discrepancy of a $[0,1]$-valued sequence $n \mapsto \alpha_n$ is the quantity $$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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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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0
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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
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0
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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
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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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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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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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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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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 $\{n\alpha\}$, $\{n^2\alpha\}$ ...
- 3,058
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0
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121
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$\delta$-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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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
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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 ...
- 2,470
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1
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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 $I\subset [0,1]$,
$$E(n,\theta, I) ={ ...
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1
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Is this a criterion for uniform distribution modulo one?
For all $k \in \mathbb N$ let $a_k$ be strictly positive bounded weights, i.e. there are constants $C_1$ and $C_2$ such that $0<C_1\le a_k \le C_2$. Now a real valued sequence $(x_k)_{k \in \mathbb ...
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0
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Convergence rate of equidistributed sequences
In the sum
$$\sum_{n=1}^N \left(\{n \alpha\}-\frac{1}{2}\right)$$
where $\{x\}$ indicates the fractional part of $x$ and $\alpha$ is an irrational number, Koksma inequality suggests an order of $\...
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The series $\sum_{n=1}^\infty {2n\brace n}^{-{2n\brace n}}$ and $\sum_{n=1}^\infty (2n)_{n}^{-(2n)_{n}}$ in the context of normal numbers
In this ocassion we consider the followgin series that involve ${n\brace k}$ the Stirling number of the second kind and $(n)_k$ the Pochhammer symbols. I've known from an informative point of view ...
1
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0
answers
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Equidistribution of linear forms over euclidean ball
Given a vector $v\in \mathbb{Z}^d\setminus\{0\}$, an irrational number $\eta$ and some big $M>0$ what type of bound can one get on $$\sum_{w\in \mathbb{Z}^d\cap B(0, M)}\exp(2\pi i \eta \cdot \...
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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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votes
1
answer
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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 ...
- 1,054
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votes
1
answer
434
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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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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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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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1
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415
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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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5
answers
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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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2
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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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