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]).