On Two Exponents of Approximation Related to a Real Number and Its Square

For each real number ξ, let λ̂2(ξ) denote the supremum of all real numbers λ such that, for each sufficiently large X , the inequalities |x0| ≤ X , |x0ξ − x1| ≤ X and |x0ξ − x2| ≤ X admit a solution in integers x0, x1 and x2 not all zero, and let ω̂2(ξ) denote the supremum of all real numbers ω such that, for each sufficiently large X , the dual inequalities |x0 + x1ξ + x2ξ| ≤ X, |x1| ≤ X and |x2| ≤ X admit a solution in integers x0, x1 and x2 not all zero. Answering a question of Y. Bugeaud and M. Laurent, we show that the exponents λ̂2(ξ) where ξ ranges through all real numbers with [Q(ξ) : Q] > 2 form a dense subset of the interval [1/2, ( √ 5 − 1)/2] while, for the same values of ξ, the dual exponents ω̂2(ξ) form a dense subset of [2, ( √ 5 + 3)/2]. Part of the proof rests on a result of V. Jarńık showing that λ̂2(ξ) = 1− ω̂2(ξ)−1 for any real number ξ with [Q(ξ) : Q] > 2.