# 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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### 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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### Show that $\lim_{N\to\infty}\int_0^1 \left|\frac1N\sum_{n=1}^N f(x+\xi_n) \right|^{\,2}\, dx = 0$ if $\int_0^1 f(x)\, dx = 0$ and $f$ is periodic

Suppose $f$ is a periodic function on $\mathbb R$ of period $1$, and $\{\xi_n\}_{n=1}^\infty$ is a sequence which is equidistributed in $[0,1)$. Prove that if $f$ is Riemann integrable on $[0,1]$ and ...
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### Does equidistribution imply convergence

The following is an interesting problem presented to this site which has yet to bet solved: Does $$\sum_{n=1}^\infty \frac{\sin(n!)}{n}$$ converge. While attempting this problem, I thought that ...
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### $\sqrt2 n^2$ is uniformly distributed. [duplicate]

How to prove $(\sqrt2 n^2)_{n\in \mathbb{N}}$ is uniformly distributed mod 1. I tried using Weyl's criterion but I seem to go nowhere. Thank You very much.
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### Strange property of irrational numbers, linked to continued fractions

Convergents of continued fractions provide in some sense the best rational approximations to an irrational number. In my research trying to prove that the binary digits of $\sqrt{2}$ are 50% one, 50% ...
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### Potential enhancement of the equidistribution theorem

The sequence $\{h_n \alpha\}$ is equidistributed mod $1$ if $\alpha$ is irrational and $h_n = n$. Is there a generalization of the equidistribution theorem with some sequence $h_n$ (other than the ...
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### If $\sqrt{h_{n+1}}-\sqrt{h_n} \rightarrow 0$ and $\alpha$ irrational, then $\{h_n \alpha\}$ is equidistributed mod 1

Here $h_n$ is a sequence of increasing integer numbers. The brackets represent the fractional part function. I am looking for a reference about this statement (which is obvious if $h_n = n$), or a ...
1 vote
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