  Let xi be a real number which is neither rational nor quadratic over Q. Based
on work of Davenport and Schmidt, Bugeaud and Laurent have shown that, for any
real number theta, there exist a constant c>0 and infinitely many non-zero
polynomials P in Z[T] of degree at most 2 such that |theta-P(xi)| < c
|P|^{-gamma} where gamma=(1+sqrt{5})/2 denotes for the golden ratio and where
the norm |P| of P stands for the largest absolute value of its coefficients. In
the present paper, we show conversely that there exists a class of
transcendental numbers xi for which the above estimates are optimal up to the
value of the constant c when one takes theta=R(xi) for a polynomial R in Z[T]
of degree d = 3, 4 or 5 but curiously not for degree d=6, even with theta = 2
xi^6.
