Simultaneous Diophantine approximation
About: This article is published in Journal of The Australian Mathematical Society. The article was published on 01 Nov 1977. and is currently open access. The article focuses on the topics: Diophantine equation & Diophantine set.
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Citations
A note on simultaneous diophantine approximation
TL;DR: This paper showed that for any two irrational numbers α, β there exist infinitely many pairs of fractions p/r, q/r satisfying the inequalities of inequalities for α and β, respectively.
13
On Optimal Constants for Best Two-Dimensional Simultaneous Diophantine Approximations
TL;DR: In this article, the main results of this paper state optimal constants for estimates of successive minima in two dimensions under a constraint on the denominator, while these inequalities are known for every dimension, best possible constants within these estimates are unknown for any dimension larger than one and remain unknown for all dimensions larger than two.
6
Simultaneous Diophantine Approximation: Searching for Analogues of Hurwitz’s Theorem
Werner Georg Nowak
- 01 Jan 2016
TL;DR: A survey on the most relevant methods and results on these constants can be found in this paper, where the authors focus on the cases of the maximum and the Euclidean norms.
5
•Posted Content
On estimating the constant of simultaneous Diophantine approximation
TL;DR: In this paper, the problem of estimating the constant of the best Diophantine approximations was studied and a lower bound of $ C_n$ for the case of n = 5 and n = 6 was given.
3
On the critical determinants of certain star bodies
TL;DR: In this paper, the authors derived a bound for the critical determinant of the star body |1|(|x1|3 + |x2|3+ |x22 + x32)3/2≤ 1.
1
References
•Book
An Introduction to the Geometry of Numbers
J. W. S. Cassels
- 01 Jan 1959
TL;DR: In this article, the authors introduce the concept of the quotient space and the notion of automorphs for diophantine approximations of diophantas in the Euclidean space.
1.4K
Simultaneous Diophantine approximation
Harold Davenport,Kurt Mahler +1 more
TL;DR: In this paper, it is shown that every irrational number admits an infinity of approximations satisfying the continued fraction expansion of the number of rational numbers pjq in relation to q for infinitely many approximation functions.
54
Simultaneous diophantine approximation
TL;DR: In this article, it was shown that the simultaneous inequalities r(p − arf < c, r(q − fir) < c have an infinity of integral solutions p, q, r (with r > 0), for arbitrary irrationals a and /3, provided that c > 1/2.6394.