The Best Approximation Function to Irrational Numbers

✍ Scribed by J.C. Tong

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 194 KB
• Volume: 49
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1994.1083

✦ Synopsis

Let (\xi) be an irrational number with simple continued fraction expansion (\xi=\left[a_{0} ; a_{1}, a_{2}, \ldots, a_{i}, \ldots\right]). Let the (i) th convergent (p_{i} / q_{i}=\left[a_{0} ; a_{1}, a_{2}, \ldots, a_{i}\right]). Let (\mu=) (\left|\left[0 ; a_{n+2}, a_{n+3}, \ldots\right]-\left[0 ; a_{n}, a_{n-1}, \ldots, a_{1}\right]\right|). In this note, we prove that among three consecutive convergents (p_{i} / q_{i}(i=n-1, n, n+1)), at least one satisfies (\left|\xi-p_{i}\right| q_{i} \mid<) (1 /\left(\sqrt{\left(a_{n+1}+\mu\right)^{2}+4} q_{i}^{2}\right)), and at least one satisfies (\left|\xi-p_{i} / q_{i}\right|>1 /\left(\sqrt{\left.\left(a_{n+1}\right)-\mu\right)^{2}+4} q_{i}^{2}\right)). The results are best possible. 1994 Academic Press, Inc.