Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

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What are some ways we measure irrationality?

I am wondering in what ways we may quantify an irrational number's approximability. This came up as I was reading about badly approximable numbers, which are those numbers $x$ such that $$\liminf_{q ...
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26 views

What is known about the approximation constant?

A badly approximable irrational is one whose continued fraction denominators are bounded; equivalently, if $\alpha$ is badly approximable then there is a $c(\alpha) > 0$ such that $$c(\alpha) = \...
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Rational approximations with denominators restricted to certain residue classes

In certain contexts in Diophantine approximation, it is important for the denominators of the convergents to be restricted to a certain residue class. Consider, for instance, this problem where it is ...
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How does the theory of Diophantine approximation change if we alter the set of numerators allowed?

The classical Diophantine approximation problem studies solutions $(p,q) \in \mathbb{Z} \times \mathbb{N}$ to inequalities roughly of the form $$\qquad \qquad \qquad \qquad \qquad \qquad \left|\alpha -...
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Concerning the fractional part function

I have that $f(x)$ and $g(x,y)$ are real polynomial with no constant terms. We have that $\forall \epsilon > 0$ the set $S_{\epsilon}$ = { $x \in\mathbb{Z} :$ Frac(f(x)) $ < \epsilon $ } is ...
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1answer
60 views

Show that an irrational square root of an integer is a badly approximable number.

Let $a\in \mathbb{Z}^{+}$ such that $\alpha=\sqrt{a}$ is irrational. Show that there is a positive number $c$ such that for every $p,q\in \mathbb{Z}^{+}$, we have, $$|q\alpha-p|>\frac{c}{q}.$$ My ...
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1answer
91 views

Is the set of “good exponents” in Diophantine approximation closed?

Given $x \in \mathbb{R}$, define its set of good exponents by $$G_{x}=\left\{\lambda \in [1, \infty) : 0<\left|x-\frac{p}{q}\right| \leq \frac{1}{q^{\lambda}} \ \text{admits infinitely many ...
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1answer
54 views

Does the set of real numbers with bounded partial quotients have positive measure?

We say a real number $x$ has bounded partial quotients if its continued fraction expansion $[a_0; a_1, a_2 \cdots]$ is bounded by some constant $M=M(x)$. The set $A$ consisting of those numbers whose ...
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2answers
41 views

What can be said about this “modified” irrationality measure?

Given $x \in \mathbb{R}$, we define its irrationality measure $\mu=\mu(x)$ by $$\mu = \inf\left\{\lambda : 0<\left|x-\frac{p}{q}\right|<\frac{1}{q^{\lambda}} \ \text{admits at most finitely many ...
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29 views

Reconciling Liouville’s theorem and Dirichlet's theorem

Liouville’s approximation theorem (at least, in the form I've seen expressed in several sources after a quick Google search, for instance this and this) states that if $\alpha$ is an algebraic number ...
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1answer
64 views

Why this rational approximation $\pi\sim\frac{80249}{25544}$ is not mentioned in OEIS?

I have checked sequence of Denominator of best approximation to $\pi$ with denominator $\le10^n$ in OEIS but I didn't find this rational $\frac{80249}{25544}$ however it is better than $\frac{22}7$, ...
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1answer
76 views

Dirichlet approximation theorem: How “sporadic” are the good denominators?

For an irrational number $\alpha \in \Bbb{R} \setminus \Bbb{Q}$, the Dirichlet approximation theorem states that there are infinitely many irreducible fractions $\frac{p}{q}$ such that $|\alpha - \...
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36 views

Prove there are no numbers with irrationality measure $ 1 < \mu < 2 $.

I know that every rational number has irrationality measure 1, while every irrational number has irrationality measure at least 2 (Dirichlet's approximation theorem). Since every number is either ...
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1answer
102 views

On Littlewood Conjecture

I started to read the following article but I stuck at the beginning: The authur say that "clearly" Eq. (1.1) holds when ... and when ... Question: Why Eq. (1.1) holds for each of the two cases ...
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3answers
105 views

Prove that the Pell's Equation $x^2 −Dy^2 = 1$ always has a solution where $y$ is a multiple of $41$

$D$ is a positive integer that is not a perfect square Recently I am taking a introductory number theory course and I met this question right after we learned Pell's equation and Diophantine ...
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18 views

Fractional parts of integral multiples of an irrational number

Fix an irrational number $\alpha \in (0,1)$. Let $\{x\}$ denote the fractional part of the real number $x$. Consider the sequence $$\{{\{\alpha}n\}:n=1,2,\cdots \}.$$ This sequence is uniformly ...
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Verifying a solution to the continued fraction representation of rational numbers

I am reading through these notes and have come across Exercise 3.5.12. I tried to basically brute force the claim with algebra using the two given equations for $\alpha$ and the recurrence relations $...
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Diophantine Approximation under certain absolute error

I'm in a discrete mathematics course right now, and we are learning about Diophantine Approximations. We covered a topic the other day that kind of went over my head. It was about finding a ...
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1answer
50 views

Asymptotic expansion for $a_n=\inf_{1\leq k \leq n}|\sin(k)|$

Can we obtain an asymptotic formula for sequences $$a_n=\inf_{1\leq k \leq n}|\sin(k)|,\ b_n=\sup_{1\leq k \leq n}|\sin(k)|$$ Moreover, what will the asymptotic formula be if we replace $\sin(x)$ by ...
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1answer
78 views

Can $q^2 \alpha \,(\text{mod } 1)$ be made arbitrarily small?

We have the standard Dirichlet approximation theorem that states that, for $\alpha \in (0,1)$, $$ \min_{1 \leqslant q \leqslant n} q \alpha \,(\text{mod } 1)\, \xrightarrow{unif.} 0 $$ uniformly in ...
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Diophantine properties of $\log(3)/\log(2)$ (and other questions)

Let's call $\gamma \in \mathbb R$ $\alpha$-badly approximable if there is some $d>0$ such that for all pairs $(k,l)\in \mathbb Z^2$ such that $ l\neq 0,$ $$ \left| \gamma - \frac kl \right| > \...
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1answer
46 views

Waring's problem for sum of squares — $\sqrt{2}$ does not lie in a minor arc

Prove that for each sufficiently large integer $N$ there do not exist positive integers $a,q$ with $a<q, (a,q)=1$ such that $q\leq N^{\frac{1}{20}}$ and the distance from $\sqrt{2} - \frac{a}{q}$ ...
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79 views

$\{mb + n \mid b\in \mathbb{R} \setminus \mathbb{Q} \text{ and } m, n \in \mathbb{Z} \}$ is dense in $\mathbb{R}$

I'm fairly new to analysis so this might be very simple. I have found other pages discussing similar questions, but I haven't found any of the answers particularly helpful for me. I've already shown ...
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1answer
40 views

How to recover a number from approximation

How can I recover a number $n$ from its approximation $a$. For example, if $n$ is $1.09861228866$, then $a$ can be $1.099$($N$ approximated value). How can I implement a function $F$ that takes $a$, ...
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1answer
38 views

Bad approximation by fractions with quadratic denominators

Are there irrational numbers $\alpha$ satisfaying that $$ \left|\alpha-\frac{m}{k^2}\right| \geq \frac{\varepsilon}{k^2} \text{ for some } \varepsilon>0 \text{ and all } k\in \mathbb{N}_{\geq 1},...
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23 views

Is this $2^{\lceil\log_2 d\rceil}$ inequality true?

Is it true that $\frac{2^{t}-1}{t}+\frac d{t}>\frac d{\lceil\log_2 d\rceil}$ holds always for every large enough $d\in\mathbb N$ and $t\geq\lceil\log_2 d\rceil$?
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25 views

Comparison between height of polynomial and height of coefficients of polynomial

Let $K$ be a number field, and let $P\in K[x_0,...,x_n],\;P=\sum_{\gamma}a_{\gamma}x^{\gamma}.$ There is a notion of the height of the polynomial, denoted $h(P)$ which can be described in one of the ...
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23 views

minimal distance between algebraic numbers of given height and degree

Let us start with an example. If $p,q$ are rational numbers of height $H$ (the maximum among the absolute value of their nominator and denominator in reduced form). Then we can bound the distance ...
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27 views

Bounding the height of an algebraic number in terms of height of the free coefficient in its minimal polynomial.

Let $K/\mathbb{Q}$ be a galois extension of degree $d$, and let $\alpha \in K$. Let's look at $\prod_{\sigma\in Gal(K,\mathbb{Q})}\sigma(\alpha)$. This is a rational number $q\in\mathbb{Q}$. Say we ...
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75 views

Roth's theorem for $m$ algebraic numbers

Roth's theorem states that Let $\alpha$ be a real algebraic number of degree $\geq 3$. Then for every $\kappa >2$ there exists a constant $c(\alpha,\kappa)>0$ such that $$|\xi -\alpha| \geq ...
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1answer
29 views

Binary recurrence sequence in $\mathbb{Z}$

Let $u_n=Au_{n-1}+Bu_{n-2}$ binary recurrence sequence in $\mathbb{Z}$ and the companion polynomial $x^2-Ax-B$ has distinct zeros $\alpha,\beta$. Then for all $n \geq 0$ $$u_n= \gamma_1 \alpha^n + \...
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1answer
38 views

Equivalent to Roth's theorem

Let $\alpha$ be a real algebraic number of degree $\geq 3$. Prove that the following three assertions are equivalent: for every $\kappa >2$ there is a constant $c(\alpha,\kappa)>0$ such ...
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1answer
40 views

proving $\big \lfloor xn-\frac 1n \big\rfloor\neq \big\lfloor xn+\frac 1n\big\rfloor$

How can I prove that for any real number $x$ there are infinitely many integers $n$ such:$$\Big \lfloor xn-\frac 1n \Big\rfloor\neq \Big\lfloor xn+\frac 1n\Big\rfloor$$
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1answer
15 views

Positive definite binary form of degree $\geq 3$, where $F(x,y)=m$ has finitely many solutions in $x,y\in\mathbb{Z}$

We have $F(x,y)\in\mathbb{Z}[X,Y]$ a positive definite binary form of degree $\geq 3$. I have to prove, without using lower bounds on linear forms in logarithms (we were working with Baker's theorems),...
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2answers
71 views

Rational solutions of quadratic Diophantine equation $ax^2+by^2+cz^2+du^2=v^2$?

What do we know about the rational solutions of quadratic Diophantine equation $ax^2+by^2+cz^2+du^2=v^2$ in five variables $x,y,z,u,v$? I am looking for references/papers related to this equation.
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178 views

Rational solutions of the system $ p^2-2 q^2 + r^2 =12, 2p^2 - 3 q^2 + s^2=42$?

I am looking for a rational solution of the system $ p^2-2 q^2 + r^2 =12, 2p^2 - 3 q^2 + s^2=42$ for $p,q,r$ with $p\ne 0$. Does it have any?
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1answer
123 views

Units in Cube Root System

I am trying to find an algorithm, similar to the Pell equation, that would solve for matricies of this form, that have a determinate of 1. ...
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62 views

Proving extension of Dirichlet 's Approximation Theorem in Analytic number theory

I am self studying Analytic number theory from Tom M Apostol and I could not think about this exercise problem in Chapter 7 . I am attaching image of problem My attempt - I tried using ...
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1answer
45 views

Bounding the norm of algebraic number of given degree and height

Let $\alpha \in \overline{\mathbb{Q}}$ of degree $\leq d$ and such that $h(\alpha)\leq h$. Denote $K=\mathbb{Q}(\alpha)$. (Here I use the definition $h(\alpha)=\frac{1}{d}\sum_{p\in spec(\mathbb{Z})}...
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443 views

In Search Of Elementary Proof Of Kobayashi's Theorem

There is a theorem in Number Theory due to Hiroshi Kobayashi (possibly less famous). The statement of this theorem is quite simple-looking. The original proof of Kobayashi relies on Siegel's Theorem ...
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1answer
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Numbers with “known” continued fractions

We know $$\frac{1+\sqrt{5}}{2} = 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cdots}}}$$ and $$e = 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\frac{1}{4+\cdots}}}}.$$ There are similar formulas for any ...
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213 views

If $m$ and $n$ are integers, show that $\left|\sqrt{3}-\frac{m}{n}\right| \ge \frac{1}{5n^{2}}$

If $m$ and $n$ are integers, show that $\biggl|\sqrt{3}-\dfrac{m}{n}\biggr| \ge \dfrac{1}{5n^{2}}$. Since $\biggl|\sqrt{3}-\dfrac{m}{n}\biggr|$ is equivalent to $\biggl|\dfrac{ \sqrt{3}n-m}{n}\biggr|...
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1answer
57 views

Rational solutions of Diophantine equation $8kx+x^4=y^2$?

Is it possible to find the rational solutions of Diophantine equation $8kx+x^4=y^2$, where $k$ is a given rational? For what values of $k$ solution exists?
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1answer
50 views

An unclear connection between two (supposedly) equivalent formulations of Liouville's approximation theorem

We shall say that $\xi\in\mathbb{R}$ is approximable by rationals to order n if there is a $K(\xi)$ for which $$|\frac pq -\xi|<\frac {K(\xi)}{q^n}$$ has infinitly many solutions. (according to the ...
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0answers
15 views

Applications of diophantine approximation in counting (combinatorics).

In what situations, if any, do Diophantine approximations of numbers come up when attempting to count something? In other words, what are some combinatorics applications of Diophantine approximations ...
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2answers
170 views

Is it true that $\lceil (3/2)^n \rceil - (3/2)^n > (3/4)^n$ for all $n>1$?

It is easy to check that $$ \lceil (3/2)^n \rceil - (3/2)^n \ge 1/2^n $$ (to see this, just look at the binary representation of $(3/2)^n$). Prove or disprove the stronger statement that $$ \lceil (3/...
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0answers
22 views

Approximating real numbers with rationals up to a constant of the real number.

Say I have a real number $x>0$ and some constant $C>0$. I want to use a rational number $q$ such that the following holds: $|x-q|<Cx$ Of course such numbers $q$ exist. What I want is to ...
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1answer
171 views

Does every $4$-dimensional lattice have a minimal system that's also a lattice basis?

An full $n$-dimensional lattice $\Lambda$ is a discrete subgroup of $\mathbb{R}^n$ (equipped with some norm $\lVert \cdot \rVert$) containing $n$ linearly independent points. If $\Lambda = \{ A z, z\...
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39 views

Successive minima

Definition: The $n$ successive minima $\lambda_1,..,\lambda_n$ of $C$ with respect to lattice $L$ are defined as follow $\lambda_i$ is the minimum of all positive reals $\lambda$ such that $\lambda C ...
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1answer
146 views

Can every algebraic integer of degree $3$ be approximated by a quotient of linearly recurrent integer sequences of degree $3$?

Given a zero $\alpha \in \mathbb{R}$ of an irreducible monic third degree polynomial $x^3 - a_2x^2 - a_1x - a_0$, are there always integer sequences $(p_n)_{n=1}^\infty$ and $(q_n)_{n=1}^\infty$ ...

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