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7 views

Probability distribution of the number of Dirichlet’s Approximations of random reals as a function of $M$

I curious about the Probability distribution of the number of Dirichlet’s Approximations of random reals as a function of $M$. What I mean by this is the following: We fix some integer $M$, and we ...
1
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1answer
33 views

Approximate irrational numbers with the same denominator

Let $\alpha$ be a irrational number, then using the continued fraction expansion we can find two sequences $\{p_n\}$ and $\{q_n\}$ with $q_n\rightarrow\infty$ as $n\rightarrow\infty$ such that ...
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0answers
19 views

Line and distance to lattice points?

Consider the lattice $\mathbb{Z}^{n}\subset\mathbb{R}^{n}$ and a line (not necessarily through the origin). What conditions can be placed on the slope of the line that is necessary and sufficient so ...
2
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0answers
26 views

Rate of convergence of an irrational rotation

Let $e^{i\phi}, e^{i\theta} \in \mathbb{S}^1$. Let $p(x) = x + 2k\pi$, where $k \in \mathbb{Z}$ is chosen so $p(x) \in [0, 2\pi)$. If we assume that $\phi/\pi$ is irrational, then there exists an ...
7
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1answer
174 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]
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0answers
24 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 $\sum_{z_x=1}^\infty qf(q)$ ...
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1answer
60 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 $x$, $$ ...
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2answers
229 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
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2answers
129 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 ...
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1answer
30 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 ...
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2answers
172 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 ...
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2answers
73 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
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0answers
24 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
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2answers
168 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
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0answers
42 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 ...
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0answers
67 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 ...
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0answers
40 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 ...
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278 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
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1answer
336 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?
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0answers
50 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 ...
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1answer
39 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
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0answers
38 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 ...
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0answers
45 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
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1answer
61 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
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1answer
43 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
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1answer
51 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
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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 ...
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58 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
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1answer
132 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
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1answer
54 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 ...
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2answers
76 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: ...
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0answers
289 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
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1answer
46 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
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2answers
247 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 ...
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5answers
716 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
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1answer
216 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 $? ...
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0answers
48 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
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2answers
97 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
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1answer
48 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.
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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 ...
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1answer
79 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 ...
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0answers
99 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
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1answer
94 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 ...
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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) = ...
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2answers
79 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 ...
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1answer
379 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
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3answers
175 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
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0answers
70 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
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0answers
51 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 ...