Mapping $\mathbb P$ onto $\mathbb Q ^\omega$

Let $\mathbb P$ denote the space of irrationals. Is there a continuous bijection (one-to-one and onto) $f:\mathbb P\to \mathbb Q ^\omega$ that maps each closed subset of $\mathbb P$ to a $G_\delta$-subset of $\mathbb Q ^\omega$? Remark 1. Suppose that $f:\mathbb P\to \mathbb Q ^\omega$ is a continuous bijection mapping closed sets to $G_{\delta}$ sets. … Read more

The constant π\pi expressed by an infinite series

I am looking for the proof of the following claim: First, define the function sgn1(n) as follows: \operatorname{sgn_1}(n)=\begin{cases} -1 \quad \text{if } n \neq 3 \text{ and } n \equiv 3 \pmod{4}\\1 \quad \text{if } n \in \{2,3\} \text{ or } n \equiv 1 \pmod{4}\end{cases} Let n=p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdot \ldots \cdot p_k^{\alpha_k} , where … Read more

Irrationality of this trigonometric function

I’d like to prove the following conjecture. Let $x = \frac{p}{q}\pi$ be a rational angle ($p,q$ integers, $q \geq 1$). Then $f(x) = \frac{2}{\pi} \arccos{\left(2\cos^4(2x)-1 \right)}$ is irrational if $x$ is not an integer multiple of $\frac{\pi}{8}$. Is this true? Is the other direction true too? I’ve tried to use the standard manipulations, Chebyshev polynomials, … Read more

Is it possible to know if $\log(\pi)$ is irrational or not since the $\log$ function is the inverse of the $\exp$ function?

I’m interested in knowing more about the question if $f(\pi)$ is rational or not, where $f$ is some well-known function. For example, $\cos(\pi) =-1$ is rational, while ${e}^{\pi}$ is irrational as shown here by WolframAlpha. My question here is: Is it possible to know if $\log(\pi)$ is rational or not since the $\log$ function is … Read more

Approximation of an irrational point from a given direction

Taking norms to be maximal norm, then the simultaneous version of the Dirichlet’s approximation theorem states that given real numbers α1,…,αd,there are integers p1,…,pd,q∈Z such that |αi−piq|≤1q1+1d. Q1 : Can we expect to control the approximation to be near a given direction, i.e. given a point P∈Rd, ϵ>0 and a unit vector e∈Sd−1, there is … Read more

Is the arithmetic-geometric mean of 1 and 2 rational?

It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, $g_{n+1} = \sqrt{a_ng_n}$ are both convergent and that $\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} g_n$. Let’s call … Read more

Irrationality of $e^{x/y}$

How to prove the following continued fraction of $e^{x/y}$ $${\displaystyle e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}}}$$ Since $a_i \geq b_i$ for all $i \geq 1$. By the condition of irrationality of generalized continued fraction, proving this directly proves that $e^{x/y}$ will be an irrational number! Answer I think this might be a solution. The … Read more

Lebesgue measure of some set of irrational numbers

Let (in) be a strictly increasing sequence of natural numbers, (vn) be an unbounded sequences of natural numbers and M≥2. Denote by I(in,vn,M) the set of all irrational numbers α=[a1,a2,…,as,…)∈(0,1) which is ain≤vn and as≤M for any s∈N∖{in,n=1,2,…}. My question related to the Lebesgue measure of this set. Of course this measure depends on exact … Read more

Irrationality of generalized continued fractions

An infinite simple continued fraction 1b1+1b2+1b3+…(bi∈Z) is irrational. Now for a generalized continued fraction: a1b1+a2b2+a3b3+…(ai,bi∈Z), the same conclusion is apparently not valid. Legendre gave a sufficient condition for irrationality: |ai|<|bi| for any i large enough. Can this result be strengthened in any way? Especially, might it hold for ai,bi∈Q? Also, can anyone give an example … Read more

Irrationality of the sum of the reciprocal of perfect powers

A couple of days ago I was trying to remember a classical exercise (which I now find out goes by the name of Goldbach-Euler theorem). Eventually I figured out that it asked to prove that ∑p∈P1p−1=1, where P={na∣n,a≥2} is the set of powers of integers. At the beginning, though, I thought that the sum I … Read more