On two exponents of approximation related to a real number and its square

Abstract

For any irrational real number xi, let lambda(xi) denote the supremum of all real numbers lambda such that, for each sufficiently large X, the inequalities |x_0| < X, |x_0*xi-x_1| < X^{-lambda} and |x_0*xi^2-x_2| < X^{-lambda} admit a solution in integers x_0, x_1 and x_2 not all zero, and let omega(xi) denote the supremum of all real numbers omega such that, for each sufficiently large X, the dual inequalities |x_0+x_1*xi+x_2*xi^2| < X^{-omega}, |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 exponents lambda(xi) where xi ranges through all irrational non-quadratic real numbers form a dense subset of the interval [1/2, (sqrt{5}-1)/2] while, for the same values of xi, the dual exponents omega(xi) form a dense subset of [2, (sqrt{5}+3)/2]. Part of the proof rests on a result of V. Jarnik showing that lambda(xi) = 1-1/omega(xi) for these real numbers xi.