For questions about approximating real numbers by rational numbers.

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3
votes
2answers
119 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 ...
3
votes
1answer
46 views

Rigorous rationale for the Pade Approximant?

I recently asked a Question for which the only Answer I got was a recommendation to punt and use the Pade Approximant. This is the first time I recalling seeing this, and intuitively it seems like ...
2
votes
1answer
27 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 ...
0
votes
1answer
30 views

Thue-Siegel-Roth Type Theorem

Dirichlet's approximation theorem says that for every real $\alpha$ and every positive integer $N$, there exist integers $p,q$ with $1 \leq q \leq N$ such that $$ |q\alpha - p| < \frac{1}{N}. $$ ...
0
votes
1answer
49 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 ...
32
votes
0answers
1k views

Irrationality of $e$

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^...
17
votes
0answers
428 views

Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{...
6
votes
0answers
108 views

An integral for $2\pi+e-9$

Motivation Lucian asked about the almost-integer $2\pi+e\approx9$ in a comment to a partially answered why question about $e\approx H_8$. This is more involved than approximations to $\pi$ and ...
5
votes
0answers
179 views

Diophantine approximation with additional constraints

I am trying to compute best rational approximations to various transcendental numbers $c$, subject to the following constraints: $$\frac {i j} {2^k} = c + \epsilon, \space\space2^n \le i, j \lt 2^{n+1}...
5
votes
0answers
76 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 ...
4
votes
0answers
34 views

Prove that there are only finitely many rational numbers $p/q$ satisfying $|\frac{p}{q}-\sqrt[d]{b}|\leq \frac{C}{q^3}$.

This is a problem from Silverman & Tate's Rational Points on Elliptic Curves. The following is the Diophantine Approximation theorem by Thue which is proved in Chapter 5: Theorem. Let $b$ be a ...
4
votes
0answers
27 views

Approximation of a rational number with values of polynomial

Given two positive integers $p,\space q$, prove that there exist a polynomial $P(x)\in\mathbb{Z}[x]$ and interval $I\subseteq\mathbb{R}$ of length $\frac{1}{q}$ such that $\forall x\in I\space\space\...
4
votes
0answers
49 views

Is there a heuristic reason behind this numerical coincidence?

Write $N(m, n; c)$ for the number of $m\times n$ zero-one matrices where each zero is adjacent to precisely $c$ others, where by "adjacent" I mean up/down/left/right but not diagonally. (Notice that ...
4
votes
0answers
153 views

Is an integer in the interval $\left[\small {3^n+1 \over 2^n+1}\lt {3^n \over 2^n} \lt{3^n-1 \over 2^n-1}\right] $ for some $\small n>1$?

Consider the expression $$ f_n(j)= {3^n+j \over 2^n+j} $$ where we select some fixed $n \gt 1$ and let j vary over the reals from +1 to -1 . I'm concerned with the problem, whether in the interval $\...
3
votes
0answers
45 views

Non-constructive proofs for the rationality of a number

One of the key ideas in transcedental number theory is proving that a number is transcedental (i.e. not the root of any polynomial with integer coefficients) by showing a sequence of rational numbers ...
3
votes
0answers
133 views

A particular Diophantine approximation of $\pi/2$.

Recently I was playing around with the sequence $$\frac{1}{n\sin(n)},\ n\in\mathbb{N}.$$ After some computations, I was led to the following question: let $p_n,q_n$ be two sequences of natural ...
3
votes
0answers
37 views

Three-gap problem, easy version.

Let $N$ be a positive integer and $\theta$ an angle in $(0, 2\pi)$. Consider the map$$f: \{0, 1, 2, \dots, N-1, N\} \to \text{unit circle}, \text{ }f(k) = k\theta \text{ }(\text{mod } 2\pi).$$Show ...
3
votes
0answers
63 views

There are infinitely many $\frac{m}{n}\in\mathbb Q$ such that $|x- \frac{m}{n} |<\frac{1}{n^2}$

Let $x$ be an irrational number. Prove that there are infinitely many rational numbers $\frac{m}{n}$ such that $$|x- \frac{m}{n} |<\frac{1}{n^2}$$ It's clear that $-1<1/n^2 <1$ with $n ...
2
votes
0answers
35 views

Average of a certain diophantine function

Yesterday I was browsing math.se and came across this question. It was answered by a few people and the best answer was already accepted so I just read the question and the solutions to it. Then I ...
2
votes
0answers
33 views

Smallest number of workers in factory, Diophantine approximation

Q. In a factory, the percentage of male workers was $53.7802\%$ (rounding to nearest fourth decimal place) last year. What is the smallest number of female workers working there? Hint: Diophantine ...
2
votes
0answers
44 views

Is there any research on Diophantine Approximation with computable numbers

I was wondering if there is any research in the field of Diophantine Approximation using the computable numbers. It seems to be a good fit, a dense countable set with a variety of different potential ...
2
votes
0answers
70 views

The exponent on Thue's theorem

I have been reading about Runge's theorem on diophantine approximation Theorem. Let $\xi$ be an algebraic real number of degree $d\geq 3$. For every $\epsilon >0$ there is a number $\gamma >0$ ...
2
votes
0answers
42 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 ...
2
votes
0answers
62 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 $d$-...
2
votes
0answers
59 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 (...
2
votes
0answers
55 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
votes
0answers
131 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 this ...
2
votes
0answers
106 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
votes
0answers
54 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 ...
2
votes
0answers
96 views

How accurately can a irrational algebraic number represent a different irrational number compared to a rational approximation?

If I am trying to approximate $x = \sqrt D$ such that D is a square free integer I can use Diophantine approximation and the Fundamental Recurrence Formulas to find a rational approximation $\dfrac{a}{...
2
votes
0answers
240 views

Diophantine equations/Diophantine Geometry

I am very knew to this site and I am eagerly waiting for solutions of: (1) Let $x$ be an algebraic number with degree $n > 1$. Then there exists only finitely many rational numbers $p/q$ (in ...
1
vote
0answers
93 views

Why does $63725\pi$ give four approximations to $\pi$?

The fraction $\frac{355}{113}$ was first used for approximating $\pi$ by the Chinese mathematician and astronomer Zu Chongzhi in the 5th century (see Milü). An isolated case? The almost-integer $113\...
1
vote
0answers
85 views

A double inequality for $\frac{\pi}{2}$

Approximating $\frac{\pi}{2}$ from above Since $$\left(\frac{\pi}{2}\right)^9\approx 58.220897$$ the root $$58^\frac{1}{9}\approx 1.5701$$ is not far from $$\frac{\pi}{2}\approx 1.570796$$ This ...
1
vote
0answers
36 views

What does $\log^3$ stand for in this paper by K. Győry?

In this paper from 1980 K. Győry proved an upper bound for the absolute value of the solutions to a given Thue-Mahler equation, but I don't understand what he meant with $\log^3$. Namely, consider a ...
1
vote
0answers
74 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 ...
1
vote
0answers
46 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 $\sum_{z_x=1}^\infty qf(q)$ ...
1
vote
0answers
80 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 ...
0
votes
0answers
33 views

Solution of diophantine equation with lowest c

Lets say I have a diophantine equation , aX - bY = c Now, for some (a,b,c) I may not have any integer solution at all. But lets say , I write the equation in this way , aX - bY = c + p p is an ...
0
votes
0answers
14 views

Recursive definition of Minkowski ?(x) function

There is a fact that $?(\frac{a+b}{c+d}) = (?(\frac {a}{c}) + ?(\frac{b}{d})) / 2$ if a/c and b/d are adjanced elements of Farey sequence. How to prove it? I don't have any ideas at all.
0
votes
0answers
18 views

Intuition in understanding Minkowski question mark ?(x) function

There are 3 definitions of Minkowski function: 1) If $[a_0; a_1, ...]$ is a continuous function representation of n 2) Consider the different ways of interpreting an infinite string of bits ...
0
votes
0answers
43 views

Prove that $p_nq_{n-1}$ - $p_{n-1}q_n=(-1)^{n-1}$ for $p_{-2}=0$ $p_{-1}=1$ $q_{-2}=1$ $q_{-1}=0$

Let $p_n$/$q_n$ for $n=0,1,2,..$ be the convergents of $a∈ R$ $p_{-2}=0$ $p_{-1}=1$ $q_{-2}=1$ $q_{-1}=0$ $p_n= a_np_{n-1}+p_{n-2}$ $q_n= a_nq_{n-1}+q_{n-2}$ I need to prove that $p_nq_{n-1} - p_{...
0
votes
0answers
54 views

On Diophantine approximation and irrationality proofs

This question is an offshoot from this previous MSE post. I have a ratio of two numbers $a$ and $b$ (presumably both positive integers), where $a$ and $b$ are determined by some arithmetic / number-...
0
votes
0answers
34 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 ...
0
votes
0answers
42 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 ...
0
votes
0answers
122 views

Solving system of equations with mixed variable types

I'm looking for solutions to the non-linear system of equations $$ n_1x + (n_1 - 1)y = a_1 \\ n_2x + (n_2 - 1)y = a_2 \\ n_3x + (n_3 - 1)y = a_3 \\ n_4x + (n_4 - 1)y = a_4 $$ where $x$ and $y$ are ...
0
votes
0answers
121 views

integer solutions to $a^m+nx^2 = y^n$ with various conditions

I consider the following equation with conditions of obtaining solutions $$a^m+nx^2 = y^n$$ This equation has solution when $a$ is an even prime and $x, y, m$ are positive integers with $(nx, y) = ...