On a Markoff-like chain (Q1073072)

The well-known Markoff chain involves two sequences \((\theta_ n)\) and \((A_ n)\), such that the inequality \(| \theta -p/q| <1/(A q^ 2)\) has an infinity of rational solutions p/q for \(A=A_ n\) and \(\theta\) not equivalent to any \(\theta_ k\), \(k\leq n\), this being no longer true for every \(A>A_ n.\) In the present paper, \(\theta\) is restricted to \(I_ r\), the class of all reals with a continued fraction expansion \([a_ 0,a_ 1,a_ 2,...]\) which satisfies \(a_ k\geq r\) from some point on (r\(\in {\mathbb{N}}\), \(r\geq 2)\). Under this constraint, the sequences \((\theta_ n^{(r)})\) and \((A_ n^{(r)})\) arising from the above approximation problem are determined and properties analogous to the classic Markoff chain are established.