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8
votes
2answers
150 views

Convergence of the sequence $\frac{1}{e^k \sin{k}}$

Does the sequence $\frac{1}{e^k \sin{k}}$ converge? If $\sin{k}$ acts as a random variable (taking on values in $(-1, 1)$), then it seems like we should be able to prove that the sequence converges ...
1
vote
2answers
64 views

Exponential irrationality

I want to prove that there are only finitely many rational solutions to $$\left|\frac{\log(5)}{\log(7)}-\frac{a}{b}\right|\le \frac{1}{7^b}$$ And once I have done this, I would like to put a bound ...
0
votes
0answers
23 views

Spatial Basis Functions…any practical hints on choosing them?

I am not coming from a pure mathematical background, and now in my daily life as an engineering student, I am getting faced to spatial basis functions to approximates variables. The problem is: Are ...
2
votes
2answers
160 views

Prove that for any give sequence of digits, there is a perfect square starting with that sequence

Prove that for any give sequence of digits, there is a perfect square starting with that sequence. With more details, prove that for $\forall a\in \mathbb{N}$, such that ...
2
votes
0answers
36 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
5
votes
0answers
60 views

finding very small values of $r + s\sqrt p + t\sqrt q$

For integers $p$, $q$ I'm interested in finding good approximations to $0$ of the form $r + s\sqrt p + t\sqrt q$ for rational $r$, $s$ and $t$. In particular, I'd like to generate approximations ...
2
votes
0answers
31 views

A generalisation of Roth's result on Diophantine approximation?

It is a celebrated result of Roth that algebraic numbers cannot be approximated by rationals too accurately: $\newcommand{\norm}[1]{\left\lVert #1 \right\rVert_{\mathbb{R}/\mathbb{T}}}$ Theorem ...
8
votes
0answers
250 views

$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news: $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can ...
14
votes
1answer
308 views

Does the sequence $\{\sin^n(n)\}$ converge?

Does the sequence $\{\sin^n(n)\}$ converge? Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge?
1
vote
0answers
47 views

Rational approximation bound for real numbers in (0,1)

I am working on a problem that related to rational approximation of real numbers. I am looking for a bound of the form: Given a positive real number, $\alpha \in (0,1)$, there exist positive ...
0
votes
1answer
34 views

Approximation of reals by rationals

Is this fact true: For all real $\alpha$, and natural numbers $n$, there exists some $a, q$ with $n^{2/3} \le q \le n$ and $(a, q) = 1$ such that $\left\lvert\alpha - \frac{a}{q}\right\rvert \le ...
0
votes
0answers
37 views

Diophantine like philosophy for computing trigonometric functions with approximation around intervals

I noticed that diophantine expressions are great to approximate constants or simple functions, as far as I know, they are not so great when it comes to approximate and compute transcendental functions ...
0
votes
0answers
29 views

Relation of n-dimensional Hausdorff measure to n-dimensional Lebesgue measure

Let $X \subseteq R^n$ then, for $s \in \mathbb{R}_{\geq 0}$ and $\rho \in \mathbb{R}^+$, we define \begin{equation} \mathcal{H}_\rho ^s (X) = \inf \left\{\sum_{i \in I} r(B_i)^s: (B_i)_{i \in I} ...
3
votes
1answer
54 views

Minkowski's theorem for non-symmetric convex bodies

Minkowski's theorem for convex bodies states that every convex, symmetric subset of $\mathbb{R}^d$ whose volume is larger than $2^d$ contains a non-zero integer point. All the proofs I've seen rely on ...
0
votes
1answer
41 views

Alternative proof of Liouville's approximation theorem

Let $\alpha\in\mathbb{R}$ be an algebraic number of degree $d\geq2$. I am asked to prove Liouville's approximation theorem using the fact that $$ \mathop{\text{den}}(\alpha)^d ...
0
votes
1answer
47 views

Rational Approximation

Suppose I have a very large integer $N$ and $a_1,....a_k$ are integers $(a_i,N)=1$ for any $1\le i\le k.$ Suppose also that $k$ is small compared to $N$ (as small as we wish). Does there exist an ...
0
votes
1answer
16 views

Equate reals by multiplication for arbitrarily large integers

How do I prove this statement? $\forall x,y>0\in\mathbb R, \forall\delta >0\exists n_1, n_2\in\mathbb N$ such that $|n_1x-n_2y|<\delta$ I have tried to prove it observing that by the ...
2
votes
1answer
24 views

approximate a vector of complex numbers

Given a vector of complex number $\vec{z}=(z_1,\cdots, z_n)$ with $|z_i|=1$ and $z_i$ is not a root of unit, and a vector of complex numbers $\vec{r}=(r_1, \cdots, r_n)$ with $|r_i|=1$. Is it the case ...
0
votes
0answers
57 views

can the power of a complex number be arbitrary close to a number?

Given a complex number $z$ with $|z|=1$ and $z$ is not a root of unit, and a complex number $r$ with $|r|=1$, and a natural number $N>0$. Is it the case that for any $\epsilon>0$, there exists ...
2
votes
1answer
113 views

Existence of a simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and for every ...
1
vote
1answer
46 views

Gap between smooth integers tends to infinity (Stoermer-type result)?

Consider the following claim : (*) Let $P$ be a finite set of primes, let $S$ be the set of natural numbers all of whose divisors are in $P$, and let $s_n$ denote the $n$-th element of $S$. Then ...
3
votes
2answers
71 views

Where can i find resourses to study this algebraic number theory?

Where can i find material to study depper the farey fractions (continued fractions)? I triying to solve problems like these: 1.- Show that two consecutive convergent at least one of them satisfy: ...
9
votes
0answers
197 views

Irrationality of $\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$

Let $\mathbb{P}$ be the set of prime numbers, and consider $m=\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$. Is $m$ irrational? In the following paper, the author recalls several sufficient ...
2
votes
1answer
42 views

Simultaneous Diophantine approximation

Is the following statement true? For all $\alpha,\beta\in \mathbb R$ and for all $\varepsilon \in \mathbb R_{>0}$, there exist $a,b,c\in \mathbb Z$ such that $|a-c\alpha|<\varepsilon$ and ...
2
votes
2answers
230 views

Does every normal number have irrationality measure $2$?

A normal number is a number whose digit expansion in any base is "uniform" in the sense that all finite digit strings occur with the "statistically expected" frequency. I read a sentence somewhere ...
0
votes
0answers
26 views

Exponential Diophantine Inequality

how would one go about solving inequality of the form $|a2^n-b2^k|>1 $ for $a,b \in R$ and $n,k \in Z$. Assume that $|a|>|b|$. Any help will be appreciated. Thank you
7
votes
5answers
695 views

Is π unusually close to 7920/2521?

EDIT: One can look at a particular type of approximation to $\pi$ based on comparing radians to degrees. If you try to approximate $\pi$ by fractions of the form $180n/(360k+1)$, you can find that ...
15
votes
1answer
196 views

Does $ \sqrt[n]{\left\lvert \sin(2^n) \right\rvert} $ have a limit?

The entire question is essentially given in the title: For $ n $ a positive integer, does the sequence $ x_n = \sqrt[n]{\left\lvert \sin(2^n) \right\rvert} $ have a limit as $ n \to \infty $? ...
2
votes
0answers
46 views

Improving a diophantine approximation

Let $x\in \mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $p/q\in\mathbb Q^d/\mathbb Z^d$ be such that $\|x-p/q\| \leqslant t$ (with $t$ small) and $q\leqslant Q$ can I say that for all $m\in\mathbb N^\star$ ...
2
votes
2answers
86 views

Criteria for irrationality of real numbers

Concerning the criteria for irrationality: Theorem says "A number $\alpha$ is irrational if and only if for every $\epsilon >0$ there exist integers $h$ and $q$ such that $0 < | q\alpha - h | ...
3
votes
1answer
45 views

Approximating the square root of two with fractions

I would like to prove that there exist only finitely many $m, n \in \mathbb{N}$ satisfying $$\left | \sqrt{2} - \frac{m}{n} \right | < \frac{1}{4n^2}.$$ Any thoughts? Thank you for your help.
13
votes
5answers
3k views

Fermat's Last Theorem near misses?

I've recently seen a video of Numberphille channel on Youtube about Fermat's Last Theorem. It talks about how there is a given "solution" for the Fermat's Last Theorem for $n>2$ in the animated ...
0
votes
1answer
66 views

How is the Borel-Cantelli lemma used in this proof on $\psi$-approximable numbers?

I'm trying to understand a paper called "Almost no points on a Cantor set are very well approximable". In the proof the author uses the Borel-Cantelli Lemma (in the eighth line at the beginning of the ...
2
votes
0answers
89 views

Diophantine approximation and Lebesgue measure

Show that the set of $x\in\mathbb{R}$ such that there are infinitely many fractions p/q with p,q relatively prime integers and $|x-p/q|\le1/q^3$ has Lebesgue measure zero. I know how to show ...
2
votes
1answer
89 views

Approximate solution of a diophantine equation

Consider the Diophantine equation $P(x)=y^2$, where $P$ is a (nonconstant) polynomial with integer coefficients and $x$ and $y$ must be integers. For $\varepsilon \gt 0$, I say that an integer $x$ is ...
0
votes
1answer
38 views

Why does a Norm Form have rational coefficients?

[From Wolfgang Schmidt's "Diophantine Approximation and Diophantine Equations"] "Let $K$ be a number field with $[K:\mathbb{Q}] = d$ and $L(X)$ be a linear form, say $L(X) = L(x_1,\dots,x_n) = ...
3
votes
2answers
71 views

Bound to the Number of Solutions of the Thue-Siegel-Roth Theorem

I have to prove for an assignment that if the partial quotients of a number $x$ are $\frac{p_n}{q_n}$, and $$\limsup\frac{\log\log (q_n)\sqrt{\log n}}{n}=\infty$$ then $x$ is transcendental. I do not ...
6
votes
1answer
231 views

Cube roots don't sum up to integer

My question looks quite obvious, but I'm looking for a strict proof for this. (At least, I assume it's true what I claim.) Why can't the sum of two cube roots of positive non-perfect cubes be an ...
2
votes
3answers
167 views

Sum of roots is integer

My question looks quite obvious, but I'm looking for a strict proof for this: Why can't the sum of two square roots of non-perfect squares be an integer? For example: $\sqrt8+\sqrt{15}$ isn't an ...
2
votes
0answers
62 views

Duffin-Schaeffer theorem/conjecture (counter)example

By the "easy" direction of Duffin-Schaeffer conjecture, it is known that if (*)$\sum_{q=1}^{\infty}\phi(q)f(q) < \infty $ (when $\phi(q)$ is euler totient function) then almost all numbers are not ...
2
votes
0answers
49 views

Least upper bound for a positive real sequence satisfying $|x_u−x_v|\cdot|u−v|>1$

The starting point of this question is the: IMO 1991, problem 6: Prove that for any $\alpha>1$, there exist a bounded real sequence $\{x_n\}_{n\in\mathbb{N}}$ such that, if $u,v$ are distinct ...
2
votes
1answer
162 views

Show that a rational number has no good rational approximations

This is homework question. The teacher proved that if $a$ is irrational, there are infinitely many rational numbers $\frac{x}{y}$ such that $|\frac{x}{y} - a|<\frac{1}{y^2}$. What we need to ...
3
votes
1answer
176 views

Kronecker's theorem - converse

I don't know how to prove that Kronecker's theorem is false if $\alpha_{1}$, $\alpha_{2}$, $\dots$, $\alpha_{M}$ are not independent. Kronecker's theorem Suppose that $\alpha_{1}$, ...
1
vote
1answer
67 views

Measure of the set of real numbers that can be approximated in this way

Let $$A = \{x \in \mathbb{R}\mid \exists\,\text{infinitely many pairs of integers $p,q$ such that $|x-p/q| \leq 1/q^3$}\}.$$ Is the measure of $A$ equal to $0$? Any ideas?
2
votes
1answer
257 views

Dirichlet approximation theorem and BA numbers

I think some version of the following definition given in "Diophantine Approximations" by W.M. Schmidt: Let $\sigma \in (0,1)$ and $x \in \mathbb{R}$. Then $x$ is $\sigma$-improvement for ...
5
votes
2answers
147 views

The real part of $z^n$

Prove that $${\displaystyle\lim \limits_{n \to +\infty}{|r^ncos(nθ)|}}=+\infty,$$ where $n$ is integer, $r>1$, $θ/π$ is irrational. I got this problem from here $1+x+\ldots+x^n$ perfect square , ...
6
votes
1answer
153 views

Finding near-integers in a range

I have a (transcendental) constant $\alpha$ and a fixed parameter $\varepsilon>0.$ I'd like to find all positive integers $n<\ell$ for which $\|n\alpha\|<\varepsilon,$ where $\|x\|$ is the ...
2
votes
1answer
42 views

Diophantine approximation problem

Denote $||x||=Min(x-[x],1-(x-[x]))$,it means the minimum distance between x and an integer. Can we find a fast algorithm to get a natural number $n$ that satisfies $$||na||<p,||nb||<p,...$$ ...
15
votes
1answer
291 views

Why is an irrational number's algebraic complexity the opposite of its Diophantine complexity?

Definition 1. Given $x \in \Bbb{R}$, the algebraic degree of $x$ is the degree of the minimal polynomial of $x$ over $\Bbb{Q}$. If $x$ is transcendental, we will define its algebraic degree to be ...
0
votes
0answers
70 views

Solving system of equations with mixed variable types

I'm looking for solutions to the non-linear system of equations $$ n_1x + (n_1 - 1)y = a_1 \\ n_2x + (n_2 - 1)y = a_2 \\ n_3x + (n_3 - 1)y = a_3 \\ n_4x + (n_4 - 1)y = a_4 $$ where $x$ and $y$ are ...