—Schmidt5 proved that for almost all real numbers α, the number of solutions in integers p, q of the inequalities
is asymptotic to a constant times log B. One might conjecture that the classical numbers (e.g., algebraic numbers, e, π) behave like almost all numbers. Machine computations’ were carried out for
... [Show full abstract] some of these numbers, and they seemed to bear out such a conjecture. Also, Lang3 has proved that the estimate is valid when α is a real quadratic irrationality.