For questions about approximating real numbers by rational numbers.

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Diophantine approximation and covers

Suppose $\alpha > 2$. Let F be the set of real numbers $x \in [0,1]$ for which the inequality $||qx|| \le q^{1-\alpha}$ is satisified by infinitely many positive integers q. For each q, let $G_q$ ...
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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 ...
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1answer
60 views

Proving that sequence elements satisfy inequality involving $\mod{1}$

I'm trying to prove that $n(3-\sqrt{8}) \; (\!\!\! \mod{1}) < m(3-\sqrt{8}) \; (\!\!\! \mod{1})\;\; \forall \; m < n \;|\;n,m \in \mathbb{N}$ $\leftrightarrow n \in ...
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1answer
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Near-integer solutions to $y=x(1+2\sqrt{2})$ given by sequence - why?

EDIT: I've asked the same basic question in its more progressed state. If that one gets answered, I'll probably accept the answer given below (although I'm uncertain of whether or not this is the ...
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2answers
79 views

Question on a constructive proof of irrationality of $\sqrt 2$

Here is the constructive proof of $\sqrt 2 \not \in \mathbb Q$ found on this page : Given positive integers $a$ and $b$, because the valuation (i.e., highest power of 2 dividing a number) of ...
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0answers
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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 ...
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0answers
35 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 ...
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Specification of Hurwitz's Theorem

Hurwitz's Theorem in Number Theory states that for every irrational number $\xi$, the equation $$\left|\xi-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}$$ has infinitely many solutions $(p,q)$ ...
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1answer
59 views

Can finite sums of two numbers come arbitrarily close to zero?

Given two real numbers $a$ and $b$, define an $a$-$b$-sum as a finite sum of $a$'s and $b$'s, i.e. a sum: $$m\cdot a + n\cdot b$$ where $m,n$ are non-negative integers. Is there a pair of numbers ...
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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 ...
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5answers
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$10^9 \times \sqrt{3}$ what are first two digits after the decimal point?

Because of floating point error, my computer basically says $10^9 \sqrt{3} \approx 1.73205 \times 10^9$ so that if we ignore the numbers before the decimal point, the fractional part is: $$\{ 10^9 ...
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1answer
35 views

Best lower bounds on difference between powers

Let $a,b\in\Bbb N$ be fixed. What are the best known lower bounds on $|a^n-b^m|$ for $n,m\in\Bbb N$, provided the difference is not $0$? If $\frac{\log a}{\log b}$ is rational, then $a,b$ are of the ...
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1answer
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Equation $x^2=y^p+1$

can you help me please for solving this dophantine equation $$x^2=y^p+1$$ and if you can give me a general method to studying such equation $$x²=y^p+t$$ Thanks
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How to find cases where $m^2$ is near to $2^A$?

In another problem here in MSE I ran into the question how I can (practically, in a program) find (positive) integer $m$ such that they are "near" to perfect powers of $2$, so $$ (0 \lt ) \qquad d_m ...
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2answers
68 views

Show that $\left| \sqrt2-\frac{h}{k} \right| \geq \frac{1}{4k^2},$ for any $k \in \mathbb{N}$ and $h \in \mathbb{Z}$.

Show that $$\left| \sqrt2-\frac{h}{k} \right| \geq \frac{1}{4k^2},$$ for any $k \in \mathbb{N}$ and $h \in \mathbb{Z}$. I tried many different ways to expand left side and estimate it but always got ...
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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} - ...
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1answer
40 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 ...
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101 views

Intuition behind proof in the Rudin book that there is no largest/smallest real number. [duplicate]

In Rudin's Principles of Mathematical Analysis (3rd ed), he proves (at the very beginning: example 1.1) that the set $A$ of all positive rationals $p$ such that $p^2<2$ contains no largest number ...
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1answer
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Diophantine approximation with integer vectors

I would like to determine whether or not there exists ${\beta > 0}$ and ${\gamma \geq 2 }$ such that ${ \forall (m_{1},m_{2}) \in \mathbb{Z}^{2} \setminus (0,0) }$, one has the inequality $$ ...
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0answers
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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 ...
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1answer
35 views

measure of $|\alpha-\frac pq|\lt\frac1{4q^2}$ with infinitely solutions

$\alpha\in[0,1]$, and $$|\alpha-\frac pq|\lt\frac1{4q^2}$$ has infinitely solutions $p, q\in\Bbb Z$, $\gcd(p,q)=1$. Let $E$ be the set of all such $\alpha\in[0,1]$, that is ...
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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 ...
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1answer
65 views

Lebesgue measure of a set of real numbers well-approximated by rationals

Consider the set $A$ of real numbers $r$ such that there exists a constant $C$ and sequence $\frac{p_n}{q_n}$ of rational numbers (where $p_n$ and $q_n$ are integers) with $q_n \rightarrow \infty$ and ...
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1answer
73 views

Approximation of irrational numbers?

Problem Suppose $\theta>1$ is an irrational algebraic integer, i.e. $\theta\not\in\mathbb Z$ but satisfies a monic polynomial with integer coefficients, and $\{a_n\}_{n\ge0}$ is a sequence of ...
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2answers
50 views

How to construct a point finite open cover of $\mathbb{R}$ that is everywhere non-locally finite?

My idea: I got the inspiration from Diophantine approximation. According to Thue–Siegel–Roth theorem, we have $$\left|\alpha-\dfrac{p}{q}\right|>\dfrac{1}{q^{2+\epsilon}}$$ for irrational $\alpha$ ...
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1answer
52 views

An uncountable and closed subset of the Liouville Numbers

I am trying to "find" a closed and uncountable subset of the Liouville's numbers. $x\in L$ means that for all $n\in \mathbb{N}$ exists $p,q\in \mathbb{Z}$ with $q>1$ such that $$0<\vert ...
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152 views

Accumulation points of $ \{x_n \in \mathbb{R}, n \in \mathbb{N} \ \ | \ x_n = n\sin(n) \}$? [duplicate]

A younger student asked me: What are accumulation points of the following set? $$ \{x_n \in \mathbb{R}, n \in \mathbb{N} \ \ | \ x_n = n\sin(n) \}$$ I really can't answer this question, could ...
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1answer
63 views

Rate of convergence of an algebraic irrational rotation

Let $\alpha \in \mathbb{S}^1$ be an algebraic number with $\mathop{\mathrm{arg}}(\alpha)/\pi$ irrational. Is it possible for the rotation by $\alpha$ to converge exponentially fast to a $\xi \in ...
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49 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 / ...
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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 ...
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1answer
105 views

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits. It is an intuitive re-statement of Baltic Way 2012 (I think there are shortlists in Baltic Way every ...
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1answer
42 views

Why can't we get a better diophantine approximation to the golden ratio?

Essentially, my question is why $|\frac{1 + \sqrt{5}}{2} - \frac{a}{b}| < 1/b^c$ (for $c>2$) is satisfied by only a finite number of $\frac{a}{b}$. This is intrinsically related to Hurwitz's ...
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59 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 $\gamma >0$ ...
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215 views

Is $e^{e^9}$ an integer?

I mean, of course $e^{e^9}$ is not an integer, but can we prove this? If you're thinking of asking Wolfram|Alpha, be warned: it gives different answers to the questions "is exp(exp(9)) an integer" (WA ...
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1answer
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How to write $\sqrt{4x^2 - 3}$ in the ring $\mathbb{Q}[x]/(x^3 - x - 1)$?

Consider the irreducible cubic equation $x^3 - x - 1 = 0$ and suppose we one of the roots $x$. The other two are $a,b$ such that $x + a + b = 0$ and $xab = 1$. Then $a$ and $b$ satisfy a quadratic ...
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Closeness of $n! \ x$ to integers for irrational $x$

This question came up in the comments to another question. Is there an irrational number $x$ such that, for sufficiently large $n$, the product $$ n! \ x $$ is arbitrarily close to an integer? More ...
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1answer
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Does there exist a quadratic generalization of the continued fraction approximants?

Let $t$ be a real number and let $\frac{p_n}{q_n}$ be its continued fraction approximants. These have the property that $$ \left| t - \frac{p_n}{q_n} \right| < \frac{1}{q_n q_{n+1}} $$ In other ...
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8answers
205 views

Rational number that approximates $\sqrt{3}$

Questions: Show that is is theoretically possible to find a rational number that approximates $\sqrt{3}$ with an error less than $0.001$. Explain how you would go about determining a ...
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On a theorem of Kronecker! [closed]

Let $\alpha$ be an irrational number and $\beta$ be an arbitrary real number, Prove that there are infinitely many pair of integers $(x,y)$ with $x\in\mathbb{N}$ such that: ...
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1answer
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Liouville's and Roth's theorems for complex algebraic numbers

Liouville's theorem says that If $\alpha$ is an irrational number which is the root of a polynomial $p$ of degree $d > 0$ with integer coefficients, then there exists a real number $C > 0$ ...
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162 views

Lower bound on $|a+b \sqrt{2} + c \sqrt{3}|$

I have stumbled across this question: Let $a,b,c$ be integers, not all $0$ such that $\max(|a|,|b|,|c|)<10^6$. Prove that $|a+b \sqrt{2} + c \sqrt{3}| > 10^{-21}$. Could anybody help by ...
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1answer
35 views

Guessing a numerical middle bound

If $x \geq 5$ and $x$ is a prime number, then what number lies between $$\frac{x + 2}{x}$$ and $$\frac{x + 3}{x}?$$ Here is my attempt: Let $\theta \in \mathbb{R}$ such that $$\frac{x + 2}{x} ...
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1answer
93 views

Thue's Theorem of Diophantine approximation

I'm trying to read the book Lecture Notes on Diophantine Analysis by Zannier and he says that the following theorems are equivalent: Theorem 1. Let $\xi$ be an algebraic real number of degree ...
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1answer
62 views

Closest rational approximation of $\sqrt x$ with denominator having prime powers $\lt n$

I am representing denominators in rational numbers with powers of their prime factors for easy multiplication and division in lowest terms (by adding and subtracting the prime powers). I would like ...
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How was this approximation of $\pi$ involving $\sqrt{5}$ arrived at?

The Wikipedia article for Approximations of $\pi$ contains this little gem: $$ \pi \approx \frac{63}{25}\times\frac{17 + 15\sqrt{5}}{7 + 15\sqrt{5}} $$ which is clearly in $\mathbb{Q[\sqrt{5}]}$. ...
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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 ...
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1answer
73 views

Approximate irrational numbers with the same denominator

Let $\alpha$ be a irrational number, then using the continued fraction expansion we can find two sequences $\{p_n\}$ and $\{q_n\}$ with $q_n\rightarrow\infty$ as $n\rightarrow\infty$ such that ...
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59 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 ...
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59 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 ...
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0answers
38 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 ...