On the condensation points of the Lagrange spectrum (Q1099211)

Let \(\alpha =[a_ 0;a_ 1,a_ 2,...]\) be the simple continued fraction expansion of \(\alpha\). If \(M=\{...,a_{-1},a_ 0,a_ 1,...\}\) is a sequence of positive integers \(a_ k\), then \[ \limsup_{k\to \pm \infty}([a_ k;a_{k+1},a_{k+2},...]+[0;a_{k-1},a_{k-2},...]) \] is called Lagrange value of M and the set of all such values is called Lagrange spectrum L. Let L(N) be the Lagrange spectrum of the Lagrange values of all sequences M, for which \(a_ k\leq N\) \((a_ k=N\) infinitely many times), where N is an arbitrary positive integer. Many authors have studied the condensation points of L and L(N). The author of the present paper proves a theorem for the constructing such points of L(N). The proof is based on a similar theorem about L, given by \textit{M. E. Gbur} [Monatsh. Math. 84, 91-108 (1977; Zbl 0372.10019)].