Location of approximations of a Markoff theorem (Q750501)

For an irrational number x with simple continued fraction expansion \([0;a_ 1,a_ 2,...]\) let \(M_ n=a_{n+1}+[0;a_ n,...,a_ 1]+[0;a_{n+2},a_{n+3},...].\) A classical result by Hurwitz and Borel asserts \[ \max \quad \{M_{n+j} | \quad j=0,1,2,\}>\sqrt{5}\quad (n\in {\mathbb{N}}). \] The first two authors [Proc. Am. Math. Soc. 97, 19-20 (1986; Zbl 0593.10030), see also \textit{E. M. Wright}, Math. Gaz. 48, 288- 289 (1964; Zbl 0163.295)] have shown that \(\max \quad \{M_{n+j} | \quad j=0,1,2\}>\sqrt{8},\) provided \(a_{n+2}=2\). As to the third minimum in the Markov chain, it is now proved that \(\max \quad \{M_{n+j} | \quad j=0,1,4\}>\sqrt{221}/5,\) if \(a_{n+2}=2,a_{n+3}=1\).