Subsequent to a theorem of Markoff

Let f(x, y) be an indefinite binary quadratic form, d(f) its discriminant, m(f) the infimum of |f(x, y)| over all integers x, y not both zero, and put $\text{μ(f) = m(f)d(f)}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$. In this paper we prove the existence of countably many disjoint open intervals I_j contained in $\text{0 ≤ x ≤}\text{1}\text{3}$ such that there is no f with μ(f) in I_j (j = 1, 2,…) and such that for any interval I containing two intervals I_j, I_k there is an f with μ(f) in I.