Given a number α which is either rational or a quadratic irrational, we can find unique integers
x,
y, and
z such that
x,
y, and
z are not all zero, the first non-zero one among them is positive, they are relatively prime, and we have :x\alpha^2 +y\alpha +z=0. If α is a quadratic irrational we can take
x,
y, and
z to be the coefficients of its
minimal polynomial. If α is rational we will have
x = 0. With these integers uniquely determined for each such α we can define the
height of α to be :H(\alpha)=\max\{|x|,|y|,|z|\}. The theorem then says that for any real number ξ which is neither rational nor a quadratic irrational, we can find infinitely many real numbers α which
are rational or quadratic irrationals and which satisfy :|\xi-\alpha| where
C is any real number satisfying
C > 160/9. While the theorem is related to
Roth's theorem, its real use lies in the fact that it is
effective, in the sense that the constant
C can be worked out for any given ξ. ==Notes==