All Questions
Tagged with diophantine-approximation irrational-numbers
8 questions
4
votes
0
answers
66
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Show that there are infinitely many well-separated grids from a fixed set of points
I have stumbled upon a question which naturally arises when trying to bin a set of $n$ points into equispaced bins such that they are sufficiently well separated from the bin edges.
Take $n$ points $...
5
votes
1
answer
586
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What is known about constructively irrational numbers?
Intuitively, a constructively irrational number is one for which we can effectively separate it from any rational number in terms of the latter's denominator. More formally, a constructively ...
- 675
1
vote
0
answers
75
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On the degree of irrationality of two irrational numbers and their rational (in)dependence
Let $x$ and $y$ be some irrational numbers. If the degree of irrationality of $x$ is the same as that of $y$, is it necessarily the case that $x$ and $y$ are rationally dependent ?
ADDENDUM: What if $...
2
votes
1
answer
162
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infinite set of mutually irrational numbers which odd linear combinations approximate 0 badly
I'm looking for a set of real numbers $\{\lambda_i;i\geq 1\}$ such that for each odd $n$, one can control $\delta_n:=\inf| \sum_i \pm n_i \lambda_i|$ where the $n_i$ are natural integers that sum to $...
- 1,352
5
votes
0
answers
109
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Approximation of an irrational point from a given direction
Taking norms to be maximal norm, then the simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}} $,there are ...
- 305
2
votes
0
answers
140
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Combination of irrationals
Fix a very small $\epsilon>0$; and irrationals $a_1,a_2>0$. Now suppose we look at all integer combinations of these irrationals which has a small norm; that is,
$$
S=\{(m_1,m_2)\in\mathbb{Z}\...
- 1,096
1
vote
0
answers
176
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binomial coefficients and irrationals
The following, probably either currently impossible to deal with, or
having a negative solution, arose from an ergodic theory question,
presumably itself currently intractible. I am not a number ...
- 4,679
2
votes
1
answer
356
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Computing all "suboptimal" rational approximations to $\pi/2$
I have an irrational number $\alpha$ ($\alpha=\frac\pi2$), and I would like to determine all integers $n\in[1,N]$ ($N=10^{16}$) that satisfy
$$ n \epsilon(n)^2 \leq \tau $$
where $\tau$ is a known ...
- 416