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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Are polynomials modulo $1$ equidistributed?
It is an elementary exercise to show that for irrational $\alpha \in \mathbb{R}$, the sequence $\{ \alpha n \mod 1 \}_{n \in \mathbb{N}}$ is dense in $T := \mathbb{R}/\mathbb{Z}$. With more work, it ...
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2answers
103 views
How to describe following set $\{ 1 \leq n \leq N: \alpha_n \in]a,b[\}$?
How to describe the set $$\{ 1 \leq n \leq N: \alpha_n \in(a,b)\}$$ when $(a,b) \subset [0,1)$ and you have following information:
a sequence of numbers $(\alpha_1,\alpha_2, \alpha_3,...)$, where ...
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1answer
25 views
uniformly fractional parts
Given $A,B$ u.d. in [0,1] and $k\in\mathbb{N}$ we define $X:=A+kB-\{A+kB\}$. How to prove $X$ is u.d..
I read something about the "Weyl equidistribution theorem" (by google). But I never heard sth ...
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2answers
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Prove $0, \frac{1}{2}, 0, \frac{1}{3}, \frac{2}{3}, 0, \frac{1}{4}, \frac{2}{4}, …$ equidistributed in $[0, 1)$
Prove
$0, \frac{1}{2}, 0, \frac{1}{3}, \frac{2}{3}, 0, \frac{1}{4}, \frac{2}{4}, ...$
equidistributed in $[0, 1)$.
A sequence of numbers $\xi_1, \xi_2, \xi_3, ...$ in $[0, 1)$ is said to be ...
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133 views
Distributing points evenly between two fixed points
I have two fixed points, $P_1$ and $P_2$.
I am trying to distribute $n$ points between them, so that the distance between every point is equal.
This is easy when:
distance between points $\cdot ...
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0answers
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Equidistribution of roots of prime cyclotomic polynomials to prime moduli
Here is a relevant - and longstanding, I'm told - conjecture. Let $f \in \mathbb{Z}[x]$ be irreducible and of degree > 1. Set
$E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { ...
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233 views
$(n^2 \alpha \bmod 1)$ is equidistributed in $\mathbb{T}^2$ if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$
I did the following homework question, can you tell me if I have it right?
We want to show that the sequence $(n^2 \alpha \bmod 1)$ is equidistributed if $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. ...
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votes
1answer
128 views
Vinogradov's equidistribution theorem
Is it true that
$(\alpha p_k)$ is equidistributed on $[0,1)$ mod 1 (Vinogradov)
$\Leftrightarrow$
$(p_k)$ is equidistributed on $[0,2\pi) $mod $2\pi$ ?
$p_k$ is the kth prime and $\alpha$ is an ...
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1answer
128 views
Is the set of logarithms of $N$-almost primes equidistributed?
Given the set of all primes. From this one builds subsets of $N$-almost primes according to a certain partition, e.g. $\lambda=(3,1)$, so the set is
$$
M=\{2^33,2^35,2^37,...,3^32,3^35,...\}.
$$
Is ...
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3answers
449 views
When is a sequence $(x_n) \subset [0,1]$ dense in $[0,1]$?
Weyl's criterion states that a sequence $(x_n) \subset [0,1]$ is equidistributed if and only if
$$\lim_{n \to \infty} \frac{1}{n}\sum_{j = 0}^{n-1}e^{2\pi i \ell x_j} = 0$$
for ever non-zero integer ...
5
votes
1answer
145 views
Theorem on behaviour of real continuous functions on the integers
I tried (and still try) to prove that $\sin (\log n)$ doesn't have a limit at $\infty$.
I know it is enough to show that a subsequence of $\log n$ approaches, modulo $2\pi$, arbitrarily close to 2 ...
15
votes
3answers
487 views
Are the fractional parts of $\log \log n!$ equidistributed or dense in $[0,1]$?
Are there any results relevant to the distribution of the sequence $\{\log \log n!\}$ for integers $n$, where $\{x\}$ denotes the fractional part of $x$?
For instance, it is known that for irrational ...
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votes
1answer
81 views
Equidistribution results vs transcendence degree
Consider $\alpha =(\alpha_1, \dots, \alpha_n) \in \mathbb{R}^n$ linearly independent over $\mathbb{Q}$, then the map $q \mapsto q \alpha:= ( q \alpha_1, \dots, q \alpha_n)$ gives a dense embedding ...
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0answers
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Distribute a given number of equally sized rectangles along a spiral path with minimum distance between adjacent rectangle edges constant
I would like to generate a set of x,y coordinates according to which thumbnails (i.e. rectangles) should appear along a spiral path so that the minimum distance between the edges of adjacent ...
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vote
1answer
125 views
Expected value of fractional part is less than 1/2
(My apologies in advance; this is very open-ended but I ask leave to post regardless.)
I'm trying to recall a theorem on the fractional part of... some fairly natural class of sequences. It showed ...
3
votes
1answer
291 views
Weyl Equidistribution Theorem and a Limit
At the moment, I'm studying Ergodic Theory and I find myself a little stuck.
The Weyl Equidistribution Theorem states that the following are equivalent:
1. For any $f \in L^{1}([0,1])$ and sequence ...
