On the Continued Fraction Expansion of a Class of Numbers

Abstract

A classical result of Dirichlet asserts that, for each real number ξ and each real X ≥ 1, there exists a pair of integers (x 0 , x1) satisfying $$ 1 \leqslant x_0 \leqslant X and \left| {x_0 \xi - x_1 } \right| \leqslant X^{ - 1} $$ (a general reference is Chapter I of [10]). If ξ is irrational, then, by letting X tend to infinity, this provides infinitely many rational numbers x 1 /x 0 with |ξ - x1/x0 ≤ x 0 - 2. By contrast, an irrational real number ξ is said to be badly approximable if there exists a constant c1 > 0 suchthat |ξ - p/q > c 1 q - 2 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 - x 1 |≤ c2X-1 admit a solution (x0, x1) ∈ ℤ2 for each sufficiently large X (see Theorem 1 of [2]).