# Tagged Questions

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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### Prove or refute that $\{p^{1/p}\}_{p\text{ prime}}$ to be equidistributed in $\mathbb{R}/\mathbb{Z}$

I've tried follow the Example 3 (see minute 30'40" of the reference), where is required the related Theorem (stated at minute 21') combined with Serre's formalism for $\mathbb{R}/\mathbb{Z}$ (also ...
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### $\{\{(2^n+3^m)\alpha\}:n,m\in\mathbb{N}\}$ is dense in [0,1]??

Let $\alpha$ be an irrational real number. I wonder whether $\{\{(2^n+3^m)\alpha\}:n,m\in\mathbb{N}\}$ is dense in [0,1] in which $\{x\}$ means the fractional part of x. This is equivalent to the ...
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### How to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]?

$\forall \alpha\in [0,1]\setminus\mathbb{Q}$, how to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]? $\{x\}$ is the fractional part of x. Any hint would be appreciated!
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### $\sum\left|\sin\alpha n+\sin\beta n\right|/n=\infty$ when $\alpha\neq-\beta$

Is there an elementary way of showing that the series $$\sum\frac{\sin\alpha n+\sin\beta n}{n}$$ is not absolutely convergent? Assuming $\alpha,\beta$ are not $0$ or $\pi$, I can show that there ...
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### Equidistribution of $\sin(n)$

It is a classical result that $$\limsup_n \sin(n) = 1$$ Even more, the set $\{\sin(n):n\in\mathbb{N} \}$ is dense in $[-1,1]$. I was wondering if it is possible to say something about the ...
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### Using Furstenberg's skew-product for $\alpha n^2$ equidistributed.

I am having trouble proving that if $\alpha_1,\alpha_2$ are irrational, then the sequence $(\alpha_1n,\alpha_2n^2)$ is equidistributed in $\mathbb{T}^2$. It is straightforward to use Furstenberg's ...
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### Can you provide us an understandable and detailed explanation of the relationship between prime numbers and equidistibution theory?

In this site, in 2. the user that answered the question wrote "the prime number theorem is an equidistribution theorem, and the Riemann Hypothesis an optimal discrepancy bound. By bounding this ...
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### Two questions about pseudo equidistributed sequences modulo 1

Let $s_n$ a sequence of positive real numbers such that $$\lim_{n\to\infty}\frac{1}{s_n}=0$$ and $$\lim_{n\to\infty}\frac{s_{[nt]}}{s_n}=t,$$ for every real $t\in[0,1]$. See here, page 4. Question ...
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### Normal numbers and equidistribution

Does anyone know how to prove the following criterion? (Due, 1949) $x \in (0,1)$ is normal for base $10$ $\iff \{10^nx\}_n$ is equidistributed. "$\Leftarrow$" is quite easy using definition of ...
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### An aquidistributed sequence

Prove that $\{an^\sigma\}$ is equidistributed in $[0,1)$,if $\sigma>0$ is noninteger and $a\neq 0$. I know how to solve this problem if $\sigma <1$ , so it is not a duplicate of ...
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### Can you think of interesting examples of equidistribtuion of a surface measure supported on a curve?

I am trying to understand equidistribution, and am trying to think of two dimensional examples where a measure is supported on a one dimensional subspace (line/curve/...), but in the limit under the ...
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Part of this blog discussing the twin prime conjecture mentions a connection between three objects: $\sum_{x \leq n \leq 2x} \tau\Big(n(n+2)\Big)$ average over twin primes where $\tau(n) = (1 \... 1answer 23 views ### Equidistribution and Smaller Sets I have a question about equidistribution. In Wikipedia, Equidistributed sequence shows in the equation in the definition that$n$reaches almost infinity. I was just wanting to make sure if ... 0answers 19 views ### Do there exist equidistributed countable subgroups in (compact) Lie groups? By an equidistributed countable subgroup I mean a countable subgroup (with a finite or possibly countable set of generators) that is dense in$G$such that for any sufficiently nice function (Haar ... 0answers 821 views ### Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best? Over in the thread "Evenly distributing n points on a sphere" this topic is touched upon: http://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere. But what I would like to ... 0answers 113 views ### Fractional part of$n\alpha$is equidistributed Let$\alpha$be an irrational number. Then the sequence$\{\{n\alpha\}\}$is equidistributed. I am using the following definition of equidistribution. A sequence$\{a_i\}$is equidistributed if$\...
I'm trying to find the number of integers $n \leq N$ such that fractional part of $\sqrt n\in (\alpha,\beta]$ where $(\alpha,\beta]\subseteq(0,1]$. The approximate number is of course $(\beta-\alpha)N$...