Tagged Questions

2
votes
1answer
36 views

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 ...
3
votes
1answer
141 views

The prime numbers do not satisfies Benford's law

A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability, $$ P(d)=\log_{10}(d+1)-\log_{10}d,$$ how do you prove that the prime numbers do ...
11
votes
1answer
138 views

Square root of an integer has only even digits

Is there a non-square positive integer $n$, that $\sqrt{n}$ has only even digits in its decimal representation ?
0
votes
1answer
60 views

The first return time of an irrational rotation

For the probability space $([0,1),\mathcal{B},\lambda)$ and for an irrational $\theta \in (0,1)$ we have the map $Tx = x + \theta \bmod 1$. I'm trying to find an expression for $n(x):=\inf\{n \in ...
2
votes
2answers
82 views

Book about ergodic theory, group actions and number theory.

Does anyone Know about an introductory book showing the intersection between ergodic theory, group actions and number theory? I have been looking for but it has been impossible to me. Thanks.
10
votes
2answers
331 views

Irrational rotation and recurrence time

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the torus, and $\alpha\in(0,1)$ be an irrational number, then the transformation $T$ defined by $Tx=x+\alpha$ is the irrational rotation on $\mathbb{T}$. ...
3
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 ...
3
votes
1answer
290 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 ...