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Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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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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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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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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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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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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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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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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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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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 ...
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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 ...
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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 ...
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$p$ 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 ...
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$0<|\sqrt a-\sqrt[3]b|<\epsilon$ 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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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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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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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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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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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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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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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 ...
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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 ...
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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} ...
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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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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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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 ...
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1answer
118 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?
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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 \...
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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 ...
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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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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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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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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 $X$, $...
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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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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 ...
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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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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
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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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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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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)-...
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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 ...
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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}...
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1answer
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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-...
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1answer
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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{...
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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}{...
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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 ($m$ -> $...
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Scope of Hurwitz's Theorem for Diophantine approximation

Hurwitz's Theorem states that for any irrational $\zeta$ there are infinitely many pairs of integers $p,q$ such that $$ |\zeta - p/q|<\frac{1}{\sqrt{5}q^2}. $$ A good Diophantine approximation $p/q$...
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A problem about effective uniformly distribution

The original problem is following: Problem 1 $\lim_{N\to \infty}|\frac{\#\{1\leq n\leq N |\ \ \ [n\sqrt2]=0(mod 2)\}}{N}|=\frac{N}{2}+O(ln(N))$. This problem is not very difficult, in fact $\#{\{...
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1answer
152 views

Dirichlet approximation theorem proof

I want to show that For all irrational numbers $a \in \mathbb{R}$ there exists an $Q \in \mathbb{N}$ such that $|a-\frac{p}{q}| \lt \frac{1}{qQ}$ for $p,q \in \mathbb{Z}$ and $1 \le q \le Q$. My ...
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662 views

show that $\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$

Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has $$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} ...
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Does Faltings's theorem imply the set of Diophantine equations are decidable?

Actually, we know all Diophantine equations are not decidable. Does Faltings's theorem imply the sets of Diophantine equations are decidable? That is , there is an algorithm that decide whether those ...