On the Continued Fraction Expansion of a Class of Numbers

(a general reference is Chapter I of [9]). If ξ is irrational, then, by letting X tend to infinity, this provides infinitely many rational numbers x1/x0 with |ξ − x1/x0| ≤ x 0 . By contrast, an irrational real number ξ is said to be badly approximable if there exists a constant c1 > 0 such that |ξ − p/q| > c1q for each p/q ∈ Q or, equivalently, if ξ has bounded partial quotients in its continued fraction expansion. Thanks to H. Davenport and W. M. Schmidt, the badly approximable real numbers can also be described as those ξ ∈ R \Q for which the result of Dirichlet can be improved in the sense that there exists a constant c2 < 1 such that the inequalities 1 ≤ x0 ≤ X and |x0ξ − x1| ≤ c2X admit a solution (x0, x1) ∈ Z for each sufficiently large X (see Theorem 1 of [2]). If ξ is rational or quadratic real, then, upon writing ξ = (qξ+ r)/p for integers p, q and r with p 6= 0 and putting c3 = |p|max{|p|, |q|}, one deduces from the result of Dirichlet that, for each X ≥ 1, there exists a point (x0, x1, x2) ∈ Z satisfying 1 ≤ x0 ≤ X, |x0ξ − x1| ≤ c3X−1 and |x0ξ − x2| ≤ c3X−1.