For questions about approximating real numbers by rational numbers.

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7
votes
1answer
195 views

Floor function inequality $\lfloor a\sqrt{2}\rfloor\lfloor b\sqrt{7}\rfloor <\lfloor ab\sqrt{14}\rfloor$

Let $a,b$ be positive integers. Show that $\lfloor a\sqrt{2}\rfloor\lfloor b\sqrt{7}\rfloor <\lfloor ab\sqrt{14}\rfloor$. [Source: Russian competition problem]
1
vote
0answers
40 views

Modification of Kinchin's Theorem

I have question regarding Kinchin's Theorem. Part of the Theorems says the following **[Kinchin's Theorem] ** Let $p,q\in \mathbb{N}$ and $a \in \mathbb{R}$. Then if $\sum_{z_x=1}^\infty qf(q)$ ...
1
vote
1answer
73 views

On some inequalities which are an generalization of F. Beukers' corresponding results

In 1981, F. Beukers proved the following theorem in his article On the generalized Ramanujan-Nagell equation I : Theorem $\bf1$. $~$ Suppose $m \in \mathbb{Z}$, then for any integer $x$, $$ ...
0
votes
2answers
266 views

How find all positive real $\beta$ such A finite number of $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

Define sequence $\{a_{n}\}$,such $$a_{1}=1,a_{2}=2,a_{k+2}=2a_{k+1}+a_{k},k\ge 1$$ Find all positive real number $\beta$,such only have a finite number of relatively prime integers $(p,q)$ ...
2
votes
2answers
166 views

How find the value $\beta$ such $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

Find all positive real number $\beta$,there are infinitely many relatively prime integers $(p,q)$ such that $$\left|\dfrac{p}{q}-\sqrt{2}\right|<\dfrac{\beta}{q^2}$$ maybe this problem background ...
0
votes
1answer
48 views

Yet another diophantine approximation question

Let $x=(x_1,\ldots,x_n)$ be a real vector in general position and let's say it is normalized: $\lVert x\rVert=1$. Let $y$ be a real number that can be arbitrarily large and $\epsilon>0$ can be ...
8
votes
2answers
204 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
86 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
32 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
185 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 ...
0
votes
2answers
64 views

How to find irrational approximates

Say I have a rational number, $n$, that approximates an irrational number of the form: $$n \approx {a+\sqrt b \over c}$$ in terms of being irrational. What is a good way of finding the unknown ...
2
votes
0answers
60 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
73 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
54 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 ...
10
votes
0answers
369 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 ...
13
votes
1answer
580 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
76 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
47 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
42 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 ...
3
votes
1answer
81 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
88 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
57 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
20 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
27 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 ...
2
votes
1answer
180 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
79 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
84 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: ...
21
votes
1answer
495 views

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

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
74 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
306 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 ...
7
votes
5answers
779 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 ...
16
votes
1answer
274 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
55 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
187 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
52 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.
19
votes
5answers
5k 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
140 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
128 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 ...
3
votes
1answer
118 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 ...
72
votes
14answers
10k views

Can the golden ratio accurately be expressed in terms of e and $\pi$

I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$ And so, I took it upon myself to try to find a way to express the golden ratio in terms of the ...
0
votes
1answer
43 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
107 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
779 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
0answers
103 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
53 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
197 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
47 views

finding rational complex numbers in a disk with least denominators

Suppose that I have a disk of radius $r$ around some complex $\alpha\in\mathbb{C}$: How would one find a complex number $g$ in that disk besides $\alpha$ such that $\mathrm{Re}(g)\in\mathbb{Q}$ ...
3
votes
1answer
346 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
124 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
397 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 ...