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

Source abstract

Abstract For each real number ξ, let denote the supremum of all real numbers λ such that, for each sufficiently large X , the inequalities | x 0 | ≤ X , | x 0 ξ – x 1 | ≤ X –λ and | x 0ξ 2 – x 2 | ≤ X –λ admit a solution in integers x 0 , x 1 and x 2 not all zero, and let denote the supremum of all real numbers ω such that, for each sufficiently large X , the dual inequalities | x 0 + x 1 ξ + x 2 ξ 2 | ≤ X –ω , | x 1 | ≤ X and | x 2 | ≤ X admit a solution in integers x 0 , x 1 and x 2 not all zero. Answering a question of Y. Bugeaud and M. Laurent, we show that the exponent where ξ ranges through all real numbers with [ℚ(ξ):ℚ] > 2 form a dense subset of the interval while, for the same values of ξ, the dual exponents form a dense subset of . Part of the proof rests on a result of V. Jarník showing that for any real number ξ with [ℚ(ξ):ℚ] > 2.