On Two Exponents of Approximation Related to a Real Number and Its Square
Damien Roy · Canadian Journal of Mathematics · 2007
Abstract For each real number ξ, let denote the supremum of all real numbers λ such that, for each sufficiently large X, the inequalities |x0| ≤ X, |x0ξ – x1| ≤ X–λ and |x0ξ2 – x2| ≤ X–λ admit a solution in integers x0, x1 and x2 not all zero, and let denote the supremum of all real numbers ω such that, for each sufficiently large X, the dual inequalities |x0 + x1ξ + x2ξ2| ≤ X–ω, |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 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.