For questions about approximating real numbers by rational numbers.
Questions tagged [diophantine-approximation]
494 questions
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How fast does $\inf_{|a|, b|, |c|\leq n}|a+b\sqrt 2+c\sqrt 3|$ go to zero?
Given a real number $\alpha$ we can define its irrationality measure $\mu=\mu(\alpha)$ as the largest number such that
$$\inf_{|a|, |b|\leq n} |a+b \alpha| \leq \frac{k}{n^{\mu+\epsilon}}$$
for some constant $k$ for all $n\in\mathbb N$ and…
Derivative
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Generalisation of Kronecker's Theorem
I am reading this [Six Lonely Runners]https://www.researchgate.net/publication/220343204_Six_Lonely_Runners, specifically Chapter 4. I do not understand their use of Kronecker's Theorem here
and the paper they cited their generalisation from is in…
crewmate
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$d^{1/3}$ badly approx. $\iff$ $d^{2/3}$ is badly approx.
Let $d$ be an integer, that is not a cube of any integer. Show that $d^{1/3}$ is badly approximable iff $d^{2/3}$ is also.
Badly approximable means that, there is a contant $C$ s.t.
$\lvert \beta-\frac pq\rvert<\frac{C}{q^2}$ holds for only…
user1161
- 659
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Badly Approximable Numbers: Lower Bound for Sum of Reciprocals of Fractional Parts
I am having a bit of trouble proving the following fact that I have seen in a couple of references on Diophantine approximation.
Let $||x|| = \min\{|x - p| ~|~ p \in \mathbb{Z}\}$. Recall that a number $\alpha$ is badly approximable if there exists…
Baldassare Romani
- 11
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Overview of "Some remarks on the Lonely Runner conjecture"
I'm reading this paper https://arxiv.org/pdf/1701.02048.pdf (Some remarks on the lonely runner conjecture - Terence Tao) and I think the maths is slightly out of my reach. Of chapter 3 I understand until midway through page 14, but I am still keen…
crewmate
- 11
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Does a limit exist for the fraction of $p\alpha_i$?
With $p$ an integer and $\alpha_i$ a real number, does $\{p\alpha_i\}\leq\epsilon$ hold for some predefined $\epsilon$, maybe depending on the number of $\alpha_i$ values? I looked at the simultaneous version of the Dirichlet's approximation theorem…
Jeroen Boschma
- 526
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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 distributed in the unit interval. However, is it the…
student
- 1,324
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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 $t\geq\lceil\log_2 d\rceil$?
Turbo
- 6,221
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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), that for each positive integer $m$ the equation…
jbuser430
- 743
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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 $1\le q\le N$ and $$|\alpha - {\frac{p}{q}}|\le…
student
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