Questions tagged [equidistribution]
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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Does half of the sequence $\left(n^{\alpha}\right)_{n=1}^{\infty}$ have even integer part/floor?
Define $e\left((a_n)_{n=1}^{\infty};N\right)$ to be the amount of members of the sequence $(a_n)_{n=1}^{\infty}$ that are $\leq N$ and have even integer part (also known as the floor of the number).
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Approximation of $1/2$ by fractional parts
Let $\{u\}$ denote the fractional part of a real $u \geq 0$, and
$\mathbb{N} = \{0, 1, 2, \dots\}$.
For positive integer $N$, define
$$
T_N(x) = \sup_{k \in \mathbb{N}} \min_{m \in [kN, (k+1)N) \cap \...
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Is $\{n\log n \pmod 1: n\in\mathbb{N} \}$ dense in $[0,1]?$ If so, is it uniformly distributed?
It is clear that $\{\log n\pmod 1: n\in\mathbb{N} \}$ is dense in $[0,1]$ but not uniformly distributed.
How about $\{n\log n \pmod 1: n\in\mathbb{N} \} ?$ Is it dense in
$[0,1]?$ If so, is it ...
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For each $n\in\mathbb{N},$ let $x_n:=\min_{1\leq k < n}\lvert\sin n-\sin k\rvert.\ $ Does $\sum_{n=1}^{\infty} x_n $ converge?
For each $n\in\mathbb{N},$ let $x_n:= \displaystyle\min_{1\leq k <
n} \lvert\sin n - \sin k\rvert.\ $ Does
$\displaystyle\sum_{n=1}^{\infty} x_n $ converge?
Consider instead, $a_1 = 0,\ a_2=1, ...
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Stuck on proof equidistribution on homogeneous space
I have been trying to understand the following proof but I don't fully understand the implicit last steps. From the replacement of the groups to the deducing of the theorems. I wondered if anyone ...
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Intuition behind Weyl's equidistribution theorem
Recently I've been studying Fourier analysis through Stein's book, and there is a section there dedicated to Weyl's Equidistribution Theorem, specifically,
A sequence $(\xi_n)_n$ in $[0, 1)$ is ...
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Questions regarding the function $\Omega(n)$
For any positive integer $n$, define $\Omega(n)$ to be the number of prime factors (including repeated factors, so for example $\Omega(12)=\Omega(2^2\times 3)=3$). It is well known (Pillai-Selberg) ...
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Question regarding behaviour of equidistributed sequences.
If $(s_1, s_2, s_3\ldots )$ is an equidistributed sequence on $[0,1],$ then for each $\ 0<\delta<\varepsilon <1\ $ and each $\ c\in [0,1-\varepsilon],\ \exists\ N\ $ such that
$$ \varepsilon -...
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$(\Delta^{(\infty)})^{p+1} \leq N(p+1)\Delta^{(p)}$ i.e. Equivalence of $p$-discrepancies
Let $\xi_1,\xi_2,\dots,\xi_N \in \mathbb{R}$, $p \geq 1$, I want to show
$(\Delta^{(\infty)}(\xi_1,\dots,\xi_N))^{p+1} \leq N(p+1)\Delta^{(p)}(\xi,\dots,\xi_N)$
Definitions :
Let $\psi(x) := \begin{...
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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 $\alpha$ ...
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General proposition that, if true, would immediately prove that $\sum\sin n$ and $\sum\cos n$ are bounded
I know that the sequence $A_n = \displaystyle\sum_{k=1}^n \sin k$ is bounded. I suspect the following general proposition is true, and it could then immediately imply $A_n$ is bounded, as an ...
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Probability that the first $m$ digits in $2^n$ are $k_1k_2\dots k_m$
I want to find the probability that the first $m$ digits of powers of 2 are a given combination $k_1k_2\dots k_m$. So far, here's my reasoning:
A number $2^n$ will have the first $m$ digits of the ...
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When is the convolution of two sequences equidistributed?
A sequence of integers $(x_n)$ is equidistributed mod $p$ if for all $a$ mod $p$, we have as $n \to \infty$:
$$ \dfrac{1}{X} \# \{n < X: x_n \equiv a \mod p \} \to \dfrac{1}{p}.$$
Let $(a_n)$ and $...
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How does this algorithm for the Van der Corput sequence work?
For any natural number $n$ write its binary expansion as $n = \sum_{i=0}^{k(n)} n_i 2^i$. Then the $n$th entry of the binary Van der Corput sequence is defined to be the dyadic rational
$$V(n) = \sum_{...
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If $x > 1$ then does the set $ \lbrace{ \lbrace{ x^n\rbrace}: n\in\mathbb{N} \rbrace}\ $ always contain either $0$ or $1$ as a limit point?
This question is somewhat related to my previous question here.
Here, $\lbrace{ \cdot \rbrace}$ means fractional part.
Is it true that if $x > 1\ $ then either $0$ or $1$ (or both) are accumulation ...
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