Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

Filter by
Sorted by
Tagged with
1
vote
0answers
16 views

What did Thue prove when, concerning rational approximations?

A follow-up to When did Liouville come up with the first transcendental numbers? In 1909, Thue showed that if $\alpha\in\mathbb R$ is algebraic of degree $n$ and $s>\frac12n+1$, and if $c$ is any ...
1
vote
0answers
22 views

Pseudo-lonely runner conjecture with $\frac{1}{k+1}$ and generalizations

I was reading about the unsolved lonely runner conjecture on Wikipedia, which states "[c]onsider $k$ runners on a circular track of unit length. At $t=0$, all runners are at the same position and ...
3
votes
3answers
138 views

Does $\min |{\cos(n)}|$ exist?

Let $f:\mathbb{Z^+}\rightarrow \mathbb{R}$ where $f(n)=|\cos(n)|$ in radians. Does $\min f$ exist? I think the answer is no and I that have the right approach to proving it. From Dirichlet ...
0
votes
0answers
22 views

Dirichlet approximation $\frac{p}{q}$ s.t. $\left| \alpha-\frac{p}{q}\right| < \frac{1}{qN}$ where $N^{\epsilon} <q <N^{1-\epsilon}$

Dirichlet's Approximation Theorem says that for all $\alpha$ and $N$ there exists $q\in [N]$ and $p$ s.t. $$ \left| \alpha-\frac{p}{q}\right| < \frac{1}{qN}. $$ This is a straighforward ...
5
votes
0answers
28 views

Chromatic polynomial of the cross-polytope and denominators of convergents to e.

Let $C_n$ denote the $1$-skeleton of the $n$-dimensional cross-polytope, and $\chi_{C_n}(x)$ be the chromatic polynomial of $C_n$. This is equivalent to the way of coloring the $(n-1)$-dimensional ...
2
votes
1answer
66 views

Finding an integer vector perpendicular to a rational vector/solving a Diophantine equation with constraints.

Essentially, I want to solve the following problem: Given a vector of rational numbers, $\vec{X}\in \mathbb{Q}^d$, find the smallest integer vector perpendicular to it. That is, find $\vec{n}\in \...
3
votes
0answers
43 views

How are almost integers of the form $(n^2-1)\pi$ related to convergents?

Two fractions related to early convergents of $\pi$ are the semi-convergents $$\frac{333-22}{106-7}=\frac{311}{99}$$ and $$\frac{355+22}{113+7}=\frac{377}{120}$$ The denominators of these fractions ...
10
votes
1answer
263 views

355/113 and small odd cubes

An important approximation to $\pi$ is given by the convergent $\frac{355}{113}$. The numerator and the denominator of this fraction are at the same distance of small consecutive odd cubes. $$\frac{...
2
votes
0answers
24 views

Effective equidistribution for number with bounded irrationality measure

This question asked for bounds greater than $\omicron\left(n\right)$ on the error $$ E_n=|T\cap\{1,2,\ldots\}|-\ell n. $$ where $$ \ell=\lim_{n\to\infty}\frac{|T\cap\{1,2,\ldots\}|}{n} \qquad\text{ ...
4
votes
1answer
47 views

Convergent subsequences of $\{ n \alpha \bmod 1 \}$

It's well known that if $\alpha \not\in \mathbb{Q}$, then the sequence $\{n \alpha \bmod 1\}_{n \geq 0}$ is dense in the torus $\mathbb{T} = \mathbb{R}/\mathbb{Z}$. Does every convergent subsequence ...
1
vote
0answers
82 views

How can one decide between structure or coincidence regarding the fine structure constant?

This question originates the following observations about the fine structure constant, $\alpha$. Measurements of $\alpha^{-1}$ yield values slightly smaller than $137.036$, and a good rational fit for ...
1
vote
0answers
35 views

Density of fractions that approximate a number

We say that a number $x\in\mathbb{R}$ is $\varepsilon$-approximated by a fraction $p/q$, with $(p,q) = 1$, if $$\Big| x-\frac{p}{q}\Big| < \frac{\varepsilon}{q^2}.$$ I'm curious about the density ...
1
vote
2answers
45 views

the second order continued fraction expansion of an irrational number and the distance to its closest integer

Let $u$ be a posive irrational number and let $$u=a_0+ \frac{1}{a_1+\frac{1}{a_2 \cdots} }$$ be its continued fraction expansion. Consider the second-order finite continued fraction expansion of $u$: ...
1
vote
1answer
125 views

How to make my algorithm work for case: irrational number $\sqrt{2}$ so that $\left | \frac{p}{q}- \sqrt{2} \right |< 0.001$

On one day, I read a magazine then I'm so interested in the algorithms, one of my favorites is RATCONVERT, i.e., if you have $\dfrac{142}{727}\cong 0.195323246..$ then how do you find $0.195323246.. \...
0
votes
1answer
45 views

Convergents of an infinite continued fraction oscillate around irrational number

If $\alpha$ is an irrational number and $\frac{p_n}{q_n}$ for $n\geqslant 0$ are the convergents to $\alpha$. Is there a way to show that the convergents alternate between being $<\alpha$ and $>...
1
vote
0answers
45 views

Minimizing a diophantine equation

If we are given $x,y$ both integers where $x>y>1$ for the expression $$(x^{\frac{a}{b}} - y)\cdot y^{b}$$ Is there any way to find integer values for a,b in which this expression is minimal and ...
0
votes
1answer
21 views

Distance to the nearest integer $<r$ implication

The following is an adaptation of a result in the middle of a proof I am reading. The result is not obvious to me and I don’t know how they were able to conclude it. I define $\langle a \rangle$ to be ...
0
votes
1answer
61 views

Distance to nearest integer inequality

If $\langle a \rangle$ denotes the distance from the real number $a$ to the nearest integer then why is it that for $a,b \in \mathbb{R}$ we have: $$min \langle a \pm b \rangle \leqslant | \langle a \...
1
vote
0answers
21 views

Proof that singular numbers are necessarily rational

A number $a \in \mathbb R$ is called singular if for all $\epsilon > 0$, there exists $Q_\epsilon$ such that for all $Q \ge Q_{\epsilon}$, there exist integers $p \in \mathbb Z$ and $q \in Z$ such ...
3
votes
1answer
71 views

How to prove $\liminf_{n \to \infty} \prod_{k=1}^n |1-\exp(2\pi k i\alpha)|=0$

I wonder how to show the following limit inferior ($\alpha$ is an irrational number) $$\liminf_{n \to \infty} \prod_{k=1}^n |1-\exp(2\pi k i\alpha)|=0.$$ Intuitively this is quite true since $\{\exp(2\...
0
votes
0answers
18 views

Roths theorem for almost all real numbers

I am trying to understand Roth's theorem for "almost all" real numbers. The statement in the theorem is that for irrational algebraic numbers, the "best" approximation exponent is ...
1
vote
1answer
34 views

A Lower Bound for Dirichlet’s Approximation

In this problem, I have successfully deduced that $\forall x\in\mathbb{R}\backslash\mathbb{Q}$, if $x$ is of the form $x=nr-[nr]$, with $n\in\mathbb{N}$ and $r\in\mathbb{R}\backslash\mathbb{Q}$ that ...
1
vote
1answer
95 views

Improving the Dirichlet's approximation theorem.

Recall that Dirichlet showed the following: For every real number $x$ and every $Q>1$, there exists an integer vector $(p,q)\in \mathbb Z^2$ such that $|xq-p|<1/Q$ and $0<q<Q$. I wonder ...
0
votes
1answer
34 views

Reference request about an irrationality measure problem

Some notations. For any real number $x$, let's define the quantity $$\mu(x):=\inf\left\{\mu\in\mathbb R_+\, \text{there is an infinity of rationals $p/q$ such that}\ \left\vert x-\frac pq\right\vert&...
0
votes
1answer
34 views

Proof of an inequality connected to Diophantine approximation

I'm recently doing some work that's vaguely connected with Dirichlet's approximation theorem. I came across this inequality that I haven't been able to prove. $$\forall\ a, b \in \mathbb{Z^+}, N\geq1,\...
4
votes
0answers
125 views

Reciprocal of a Liouville number is also a Liouville number

Prove that the reciprocal of a Liouville number is also a Liouville number I am using the definition of a Liouville number given in the book Transcendental Numbers by M. Ram Murty. A screenshot of ...
6
votes
1answer
139 views

G.Rhin's lower bounds for $ | S \log 2 - N \log 3 |$ (used by J. Simons in the Collatz-problem) adaptable to $ | S \log 2 - N \log 5 |$?

By G. Rhin, cited by John Simons, 2007, we have the upper bound for $$ |S \log2 - N \log 3 | \gt \exp(-13.3(0.46057+\log(N))) \qquad \text{roughly:} {1\over 457 N^{13.3}} $$ This has been used by John ...
0
votes
0answers
57 views

When a rational series converges to an irrational number, can I operate on the sum of the rational series as I would a rational number?

I am working on a problem involving Diophantine approximation. For some irrational value $0<\alpha<1$, I construct an infinite series $S = x + \frac{x_1}{10} + \frac{x_2}{10^2} + \frac{x_2}{10^3}...
1
vote
1answer
59 views

How to understand this step in proof of Kronecker's theorem

I was reading a proof of Kronecker's theorem, which is: Let $\theta$ be an irrational number. For all real $\alpha$ and all $\epsilon >0 $, there exist integers $a,c$ with $|a\theta - \alpha - c| &...
0
votes
0answers
29 views

Bound over finite and infinite places

I'm looking at a proof of Dobrowolski Theorem by Amoroso and I don't quite get why we have the following bounds for a polynomial $F \in \mathbb{Z}[x]$ of degree $L$ vanishing on $\alpha$. $\left|F\...
4
votes
3answers
114 views

Why do all integer solutions of $\frac{157}{50}<\frac{p}{q}<\frac{22}{7}$ for $q>0$ have the form $p=157n_1+22n_2+179,q=50n_1+7n_2+57$?

This is from my attempt to show $\frac{22}{7}$ is the best approximation of $\pi$ with denominator not more than $50$. My attempt as follows: Suppose there is a better rational approximation, $\frac ...
6
votes
6answers
246 views

Is $22/7$ the closest to $\pi$, among fractions of denominator at most $50$?

Is $22/7$ the closest to $\pi$, among fractions of denominator at most $50$? I am currently studying continued fractions, while I know that for all denominators at most $Q_n$, $\frac{P_n}{Q_n}$ is the ...
29
votes
3answers
1k views

Is the sequence $(B_n)_{n \in \Bbb{N}}$ unbounded, where $B_n := \sum_{k=1}^n\mathrm{sgn}(\sin(k))$?

This question is kind of an extension of a previous question I asked here. The infinite series $$\sum\frac{\mathrm{sgn}(\sin(n))}{n}$$ does converge, but I would like to know if Dirichlet's test can ...
2
votes
0answers
28 views

Compute height of points of a jacobian variety using Mumford representation

Let $C$ be a hyperelliptic curve, we know its jacobian (or $Pic^{0}C$) is in one to one correspondence to reduced divisors. In the reduced divisor representation, any class is expressed in the form $[...
1
vote
1answer
58 views

Another proof of $\displaystyle\limsup_{n\to\infty}|\cos{n}|=1$

I have seen a proof of $\displaystyle\limsup_{n\to\infty}|\cos{n}|=1$ by using density of $\{a+b\alpha: a,b\in \mathbb{Z}\}$ in $R$, where $\alpha$ is irrational. Here I give another proof of as ...
2
votes
2answers
144 views

Some question about proving $\displaystyle\limsup_{n\to\infty}|\cos{n}|=1$ by using density of $\{a+b\pi|a,b\in\mathbb{Z}\}$

I have seen Proving $\displaystyle\limsup_{n\to\infty}\cos{n}=1$ using $\{a+b\pi|a,b\in\mathbb{Z}\}$ is dense and got this question. Hagen von Eitzen gave the solution as following: Pick an integer $n$...
3
votes
2answers
80 views

Prove there are finitely many pairs of integer x, y such that $|x-\sqrt{d}y|<\frac{1}{y^2}$

Prove there are finitely many pairs of integer x, y such that $|x-\sqrt{d}y|<\frac{1}{y^2}$ where $d$ is a non-square natural number. I know there're infinitely many pairs of integer x, y such that ...
0
votes
0answers
65 views

Prerequisites for Diophantine Approximation

I’m currently reading Fractal Geometry by Falconer to help learn some diophantine approximation theory, but would like to know if there are any supplementary texts which focus more specifically on ...
1
vote
0answers
43 views

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 ...
1
vote
0answers
34 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) = \...
3
votes
0answers
27 views

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 -...
0
votes
0answers
36 views

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 ...
3
votes
1answer
100 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 ...
2
votes
1answer
98 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 ...
5
votes
1answer
89 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 ...
2
votes
2answers
43 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 ...
0
votes
0answers
41 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 ...
2
votes
1answer
158 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 $\frac{22}7$, ...
2
votes
1answer
103 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 - \...
0
votes
1answer
116 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 ...

1
2 3 4 5
8