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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Equidistribution of $an^\sigma$

I am stuck at an exercise in Stein's book: Fourier Analysis, and it's exercise 8 in chapter 4. Show that for any $a\ne0$, and $\sigma$ with $0<\sigma<1$, the sequence $an^\sigma$ is ...
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Functional equation relating to normal numbers

My coauthor and I have run into the following problem in a research project involving normal numbers. We suspect that the following question may be resolved using standard techniques in analysis. We ...
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Prove that eventually Hannah and the other swimmer will settle into a pattern where they pass each other (Please refer to the context in my question)

From the 2014 Mathcamp quiz: Hannah is about to get into a swimming pool in which every lane already has one swimmer in it. Hannah wants to choose a lane in which she would have to encounter the other ...
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Equidistribution of lattice points on spheres in dimensions $d \ge 4$ (Pommerenke's theorem)

I am looking for either an English translation of: Pommerenke, C.: Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden. Acta Arith. 5, 227-257 (1959) Or a reference in ...
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Generalization of the Riemann integral to functions on the sphere

I want to do something a bit strange: define the Riemann/Darboux integral for bounded real functions defined on the n-dimensional sphere. The catch: I cannot use Lebesgue integration theory to do ...
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Algorithms for non-random but equidistributed ways to fill up a Cartesian plane

In pages 90-91 of this book the authors talk about uniform, but not necessarily normally distributed random ways to fill up a Cartesian grid. For example, in the attached images. These are the ...
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Equidistribution of $\{\xi_n\}$ where $\xi_n = <n\frac{p}{q}>$ $p,q$ rel. prime

I'm working from Stein's An Introduction to Fourier Analysis, and there's a question (chapter 4 number 6): Let $\theta = \frac{p}{q} \in \mathbb{Q}$ where $\operatorname{gcd}(p,q) = 1$. Assume ...
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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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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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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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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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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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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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$(n^2 \alpha \bmod 1)$ 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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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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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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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 ...
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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 ...
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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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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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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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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 ...
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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 ...