Journal Article10.1112/PLMS/S3-2.1.406
Simultaneous Diophantine Approximation
19
TL;DR: It is well known that every irrational number possesses an infinity of rational approximations p/q satisfyingIt is also known that there is a wide class of irrational numbers which admit of no approximation which are essentially better, namely those θ whose continued fractions have bounded partial quotients as discussed by the authors.
read more
Abstract: It is well known that every irrational number θ possesses an infinity of rational approximations p/q satisfyingIt is also well known that there is a wide class of irrational numbers which admit of no approximations which are essentially better, namely those θ whose continued fractions have bounded partial quotients. For any such θ there is a positive number c such that all rational approximations satisfy
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
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.
The two-dimensional Diophantine approximation constant. II.
TL;DR: In this article, it was shown that the suprema of cx(a, β) and c2(α, β), taken over all α, β such that 1, α,β is an integral basis for a real cubic field, are equal, and a necessary and sufficient condition for this common value to be equal to 2/7 is given.
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