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

Damien Roy

Published 2004-09-14, updated 2004-10-05Version 2

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.