We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
A Determinantal Approach to Irrationality.
- Authors
Zudilin, Wadim
- Abstract
It is a classical fact that the irrationality of a number $$\xi \in \mathbb R$$ follows from the existence of a sequence $$p_n/q_n$$ with integral $$p_n$$ and $$q_n$$ such that $$q_n\xi -p_n\ne 0$$ for all n and $$q_n\xi -p_n\rightarrow 0$$ as $$n\rightarrow \infty $$ . In this paper, we give an extension of this criterion in the case when the sequence possesses an additional structure; in particular, the requirement $$q_n\xi -p_n\rightarrow 0$$ is weakened. Some applications are given, including a new proof of the irrationality of $$\pi $$ . Finally, we discuss analytical obstructions to extend the new irrationality criterion further and speculate about some mathematical constants whose irrationality is still to be established.
- Subjects
IRRATIONAL numbers; MATHEMATICAL sequences; OBSTRUCTION theory; MATHEMATICAL constants; CHEBYSHEV approximation
- Publication
Constructive Approximation, 2017, Vol 45, Issue 2, p301
- ISSN
0176-4276
- Publication type
Article
- DOI
10.1007/s00365-016-9333-7