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Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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1answer
103 views

Does $\sum_{k=1}^n|\cot \sqrt2\pi k|$ tends to $An\ln n$ as $n\to\infty$?

Question: How can we prove that $$L(n)=\sum_{k=1}^n\left|\cot \sqrt2\pi k\right|=\Theta(n\log n)$$ as $n\to\infty$? Furthermore, if $\sqrt2$ is replaced with a quadratic irrational number, does it ...
11
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2answers
179 views

Convergence of $\sum_{n=1}^\infty\frac{\cot \varphi\pi n}{n^s}$

Question: $s\in\mathbb C$, Is $$\sum_{n=1}^\infty\frac{\cot \varphi\pi n}{n^s}$$ absolutely convergent, conditionally convergent or divergent, where $\varphi=\frac{1+\sqrt5}2$? TL;DR, my progress ...
6
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1answer
165 views

Inequality $|\cos(k)| \geq \frac{1}{2^k}$ for $k\geq 0$

My question : Is it true that $|\cos(k)| \geq \frac{1}{2^k}$ for all integers $k\geq 0$ ? What I tried : I have checked with a computer that the inequality holds for $0 \leq k \leq 4\times 10^5$. I ...
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0answers
8 views

Citable reference (book) for a variant of Kronecker's Approximation Theorem

Let $\tau,\sigma\in(0,\infty)$ with $\frac{\tau}{\sigma}\notin\mathbb Q$. A common version of Kronecker's Approximation Theorem is the following: For each $x\in \mathbb R$ and $\epsilon>0$, ...
33
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1answer
792 views

Examples of transcendental functions giving almost integers

Informally speaking, an "almost integer" is a real number very close to an integer. There are some known ways to construct such examples in a systematic way. One is through the use of certain ...
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0answers
197 views

An Engineer sets out to Prove Fermat's Last Theorem …

This started off as a joke post of mine on a Facebook Group called "Bad Maths that Gives the Right Answer", in which I pulled a Fermat and claimed that the last bit of the proof was too long to post. ...
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0answers
28 views

Quasi-periodic sequence

Let $f(\theta)$ be some $2\pi$-periodic function which takes the values $f(\theta) \in \{1,-1\}$. Further let $Q$ be some number which is rationally independent of $2\pi$ (More specifically take $Q/(2\...
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2answers
189 views

A variant of Kronecker's approximation theorem?

Let $\tau,\sigma\in(0,\infty)$ with $\frac{\tau}{\sigma}\notin\mathbb Q$. By Kronecker's approximation theorem, we know: (1) For each $x\in \mathbb R$ and $\epsilon>0$, there are $m,n\in\mathbb ...
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1answer
29 views

Existence of interesting infinite subsets of the rationals whose elements are not arbitrarily close to some reals

It is well known that for every real number we can find a rational number arbitrarily close to it. In other words: Let $a\in\mathbb{R}$. For each $\epsilon\gt 0$, there exist $p,q\in\mathbb{Z}$ such ...
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2answers
190 views

What is $T^1(\mathbb H^2/PSL_2(\mathbb Z))$?

Let $\mathbb H^2$ be the upper-half plane. The group $PSL_2(Z)$ acts on $\mathbb H^2$ by isometries, and hence we get an action on $T^1(\mathbb H^2)$. This action is free, smooth, and proper, and thus ...
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27 views

What is sum of squared elements of Farey sequence?

Let's consider a Farey sequence: $$F_{n}=\{a_{1},...,a_{k}\}$$; Where given elements satisfy definition of $n$-th Farey sequence. My problem: Find the formula for the following sum: $$\sum_{l=1}^{k}...
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1answer
38 views

Exercise 1.12 from Ed Burger's book The Number Jungle.

An earlier exercise asks for a proof of the following result: Corollary 1.9 Let $\alpha$ be a real number and $N$ a positive integer. Then there exists a rational number $p/q$ such that $1\le q\le N$ ...
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2answers
54 views

Bad approximation, alternative definiton

I read Complex dynamics of Carleson and Gamelin. They state without a proof the following $\theta$ is called bad approximate if there exists $c>0$ and $\mu<\infty$ such that $$ \Big\vert \...
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28 views

Diophantine Approximation on Quadratic Polynomials

Given an integer $a$ which is not a perfect square, I'd like to ask how to perform Diophantine Approximation of $\frac{x^2}{y^2}$ to $a$ where $x$ and $y$ are integers. Specifically, integers ...
0
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1answer
32 views

Equidistribution of $\{p_n^2 \alpha \}$

Let $p_n$ be the $n$th prime and $\alpha$ an irrational number. Vinogradov proved that the sequence $\{p_n \alpha \}$ is equidistributed. Is it known whether the sequence $\{p_n^2 \alpha \}$ is ...
14
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2answers
532 views

Prove $\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}$

$$\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}.$$ Prove it converges and, evaluate the series. For the first part of the question, I prove it ...
2
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2answers
60 views

$p$ is adherent value of $\left(\sum_{i=1}^{p} z_{i}^n\right)_{n\in\mathbb{N}}$ where $z_{i}$ are complex of modulus 1

Here is an exercise I'm trying to solve : Let $z_{1}, \ldots,z_{p}$ be some complex numbers of modulus 1 and, for $n\in\mathbb{N}$, $u_n = \sum\limits_{i=1}^{p} z_{i}^n$. Show that $p$ is adherent ...
9
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1answer
105 views

$0<|\sqrt a-\sqrt[3]b|<\epsilon$ for $a,b\in\Bbb Z_+$

I'm trying to solve the following problem: Given $\epsilon>0$, are there positive integers $a,b$ such that $0<|\sqrt a-\sqrt[3]b|<\epsilon$ ? My solution: given $n\in\Bbb N$, $$|\sqrt{n^2}...
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0answers
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Irrationality measure of $\frac{\ln{3}}{\ln{2}}$.

Is there anyone knows the irrationality measure of $\frac{\ln{3}}{\ln{2}}$? Since it's transcendental number, its measure is $2$ or greater, and I have not found something relevant. Thanks a lot!
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1answer
66 views

Does $lcm\{1,2,…,n\} = \prod_{p\leq n, p\in\mathbb{P}}p^{\lceil \frac{log(n)}{log(p)}\rceil}$?

I am trying to understand Apery's proof of the irrationality of $\zeta(3)$ from start to end, with this document. I apologise for having 2 questions in one, but both are relatively simple (I just need ...
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0answers
26 views

An exponent for a linear homogeneous diophantine approximation

Notations. Let's denote by $\vert \cdot\vert $ the standard euclidean norm on $\mathbb Z^2$. Let's denote by $\Vert\cdot\Vert_{\mathbb Z}$ the following norm on $\mathbb R^2$: $$\forall X\in\mathbb ...
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0answers
36 views

Siegel's lemma for arbitary number field.

I want to show a proposition related to Siegel's lemma: Let $K$ be a number field of degree $d$, $L_j = \sum_{i} a_{ij}X_i \in \mathcal{O}_K[X_1, \cdots, X_N] (j= 1, \cdots, M)$ linear forms in $K$ ...
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0answers
72 views

Approximating a real arbitrarily well by $Z$-linear combination of two reals having irrational ratio.

$\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\Z}{\mathbb Z}$ $\newcommand{\R}{\mathbf R}$ Let $\alpha$ and $\beta$ be positive real numbers such that $\alpha/\beta$ is irrational. Then the following ...
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99 views

How many integer solutions are there to $x^4+y^4-x^2y^2=n$. Is there a generating function for this?

It would be kind of cool to get a closed form for the number of integer solutions $$x^4+y^4-x^2y^2=n$$ which we will let $\phi_n$ denote. It would be cool because we could exploit $\sum_{n=1}^N\...
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0answers
92 views

Finding a minima for a linear form with integer coefficients

Some context. This question is aiming to fill gaps in a larger proof, so in a way, it is kind of related to this two other questions (this one and that one) that I asked earlier. But since the ...
3
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2answers
116 views

A linear form can not be too small on integer points

Notations. Let $\xi=(\xi_1,\xi_2,\xi_3,\xi_4)\in\mathbb( R\setminus\mathbb Q)^4$ such that $\xi_1\xi_4-\xi_2\xi_3\ne 0$ and the $\xi_i$ are linearly independent over $\mathbb Q$. I have the ...
4
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1answer
123 views

Finding the minimal value of a $4\times 4$ determinant

The question. Let $\xi=(\xi_1,\xi_2,\xi_3,\xi_4)\in\mathbb R^4$ be a vector with irrational coordinates. I am interested in finding the minimal value $\mu_\xi$ of $$\left\vert \det \begin{pmatrix} ...
2
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2answers
56 views

Why we get inequality form an equation?

In the paper linear forms in the logarithms of real algebraic numbers close to 1, it is written on page 5 that- $\varLambda \leq \frac{1}{by^n}$ (see equation 7 on page 5) But we get it from an ...
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0answers
43 views

Motivation behind proof of Behrend's theorem on size of AP free subset

Behrend's theorem regarding AP states that There exists an absolute constant $c$ such that for all sufficiently large integers $N$ there exists a subset $A$ of $\{1, 2, \cdots, N \}$ with at ...
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2answers
478 views

Exploiting a Diophantine approximation of $\pi^4$ into giving a series of rationals for $\pi^4$

A note about this question: The original question asked seems likely impossible so I am really asking if we can exploit the technique below into giving us a 'nice' form for $\pi^4$. By nice form I ...
2
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1answer
120 views

Any suggestions on how to compute $\limsup |\cos n|^{n^2}$?

This problem has proven very difficult, does anyone have any suggestions on how to tackle it? Any little known theorems/identities that might help?
2
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3answers
233 views

Show that $x^2 + y^2 = 3$ has no rational points [duplicate]

Are there rational numbers such that $x^2 + y^2 = 3$ ? If I want to find a rational paramterizatio of $x^2 + y^2 = 1$ could start with the point $(1,0)$ and find lines $\ell$ of slope $m \in \...
4
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1answer
64 views

Can we remove the absolute value when doing Diophantine Approximation?

This is a very general question. For Diophantine Approximation propositions, the statements always include the absolute value sign, but I think if we can remove the absolute value sign, the ...
2
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2answers
92 views

Evaluation of infinite series of ratios (with denominators given as finite products) in terms of rational numbers and $\pi\sqrt3$

How can we prove whether $$\sum_{k=0}^\infty \frac{1}{\prod_{i=1}^{6n+2}(3k+i)}=q_1+q_2\sqrt{3}\pi$$ for all natural $n$ with rational $q_1$ and $q_2$? Some series related to rational ...
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0answers
61 views

Is $n^p +\tan n,p>0$ always no no lower bound?

I know that $\tan n,n\in\mathbb{Z}_+$ is dense on $\mathbb{R}$, so there's no lower bound for $\tan n$. But what if add a positive sequence $n^p,p\geq0,n\in\mathbb{Z}_+$ that increased faster than $n\...
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4answers
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Does the sequence $n+\tan(n), n \in\mathbb{N}$ have a lower bound?

Is the sequence $n+\tan(n), n \in\mathbb{N}$ bounded below? Intuitively I think it is not bounded below, but I have no idea how to prove it. It is like a Diophantine approximation problem, but most ...
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vote
1answer
47 views

Inequality with sum involving minimums

I am working through lemma 2.2 of the Hardy-Littlewood Circle Method: Second Edition by R.C. Vaughan and I am having some trouble with a step in his proof of the following lemma: Suppose that $X$, $...
4
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1answer
129 views

exponentiating the natural numbers

I'm not very knowledgeable on number theory, but the other day, I was thinking about this problem: Given any integer number $N>0$ which is not a power of $10$, there exists a positive integer $...
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0answers
38 views

Determine all limits of subsequences of $|\lambda \alpha^n +\mu \bar{\alpha}^n|$ (assume $|\alpha|>1$)

Let $\alpha,\lambda,\mu$ be nonzero complex numbers with $|\alpha| \gt 1$, and let $u_n=|\lambda \alpha^n + \mu \bar{\alpha}^n|$. My question: Can anyone determine the set of all so-called adherence ...
2
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1answer
43 views

Continued fraction rate of convergence

I am reading the Wolfram page for 'convergents' (of continued fractions) and it states without proof the following inequality $$\frac{1}{(a_{n+1}+2)q_n^2}<|\alpha-\frac{p_n}{q_n}|<\frac{1}{a_{n+...
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0answers
99 views

Diophantine condition of irrational numbers

A vector $\xi\in \mathbb{R}^n$ satisfies a Diophantine condition if there exists a constant $X_\xi = C(\xi)>0$ and $\sigma>0$ such that \begin{equation}\label{Diophantine condition} |\xi\cdot \...
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1answer
88 views

Dirichlet's Approximation Theorem for integer

This theorem says that if $\alpha$ is a real number and $n$ is a positive integer, then there exist integers $a$ and $b$ with $1\leqslant a\leqslant n$ such that $\vert a\alpha - b\vert < 1/n$. ...
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93 views

Lagrange spectrum in diophantine approximation theory

Context. Hurwitz' theorem states that for every irrational $\xi$, there is infinitely many rationals $p/q$ such that $$\left\vert \xi-\frac pq\right\vert<\frac 1{q^2\sqrt 5}.$$ The number $\sqrt ...
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0answers
180 views

Can interesting bounds to Gauss circle problem be seen/come from counting points close to a line?

Let $N(r)$ be the number of lattice points distance at most $r$ from the origin in $\mathbb{R}$. The Gauss circle problem is a famous problem which is looking to understand the error term $E(r):= N(r)-...
2
votes
2answers
133 views

Dirichlet's approximation theorem

A corollary of Dirichlet's approximation theorem is that for any irrational $\alpha$, there are infinitely many integer solutions of $$\left|\frac{p}{q}-\alpha\right|<\frac{1}{q^2}$$ Are there any ...
3
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0answers
61 views

Fractional part of rational powers

So I know that given some squarefree integer $n$, the fractional part of $\sqrt{n}$ can be bounded below by $\frac{1}{2\sqrt{n}+1}$ by $$n=\left[\sqrt{n}\right]^2+2\left[\sqrt{n}\right]\left\{\sqrt{n}...
3
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1answer
52 views

Real part of complex root of $X^5-X-k^5+k+1$ is $\frac{1}{23k}$ away from any integer ($k\geq 2$)

Let $k\geq 2$ be an integer. The polynomial $P_k=X^5-X-k^5+k+1$ is easily seen to have exactly one real root and two pair of conjugate non-real roots. Is it true that if $\alpha_k + i \beta_k$ any non-...
1
vote
1answer
45 views

Find $\frac{a}{b} \in \mathbb{Q}$ such that $ |\,\frac{a}{b} - \sqrt{2}|_3 < \epsilon $

Are there methods for approximating square roots of numbers in two different $p$-adic valuations. The squares mod $3$ are $\square =\{ 0,1\}$. Therefore, by Hensel's lemma $\sqrt{2} \notin \mathbb{...
4
votes
1answer
151 views

Intuition behind the irrationality measure

The irrationality measure $\mu(x)$ of a real number $x$ is defined to be the supremum of the set of real numbers $\mu$ such that the inequalities $$0 < \left| x - \frac{p}{q} \right| < \frac{1}{...
10
votes
3answers
340 views

M and n are positive integers such that $2^n - 3^m > 0$. Prove (or disprove) that $2^n - 3^m \geqslant 2^{n-m}-1$.

Given that $2^n - 3^m > 0$, I know that $n > m\log_{2}3$ (*). If $2^n - 3^m \geqslant 2^{n-m}-1$, $n>= m + \log_{2}\frac{3^m-1}{2^m-1}$ (**). This is the result when I graph it out ($m$ -> $...