214 IV Continued Fractions and Their Uses
A complex numberζis said to be analgebraic number,orsimplyalgebraic, of
degree dif it is a root of a polynomial of degreedwith rational coefficients which is
irreducible over the rational fieldQ. Thus an algebraic number of degree 2 is just a
quadratic irrational.
For any irrational numberξ, there exist infinitely many rational numbersp/qsuch
that
|ξ−p/q|< 1 /q^2 ,since the inequality is satisfied by any convergent ofξ. It was shown by Roth (1955)
that ifξis a real algebraic number of degreed≥2 then, for any givenε>0, there
exist only finitely many rational numbersp/qwithq>0suchthat
|ξ−p/q|< 1 /q^2 +ε.The proof does not provide a bound for the magnitude of the rational numbers which
satisfy the inequality, but it does provide a bound for their number. Roth’s result was
the culmination of a line of research that was begun by Thue (1909), and further
developed by Siegel (1921) and Dyson (1947).
A sharpening of Roth’s result has beenconjecturedby Lang (1965): ifξis a real
algebraic number of degreed≥2 then, for any givenε>0, there exist only finitely
many rational numbersp/qwithq>1suchthat
|ξ−p/q|< 1 /q^2 (logq)^1 +ε.An even stronger sharpening has been conjectured by P.M. Wong (1989) in which
(logq)^1 +εis replaced by(logq)(log logq)^1 +εwithq>2.
For real algebraic numbers of degree 2 we already know more than this. For, ifξis a
real quadratic irrational, its partial quotients are bounded and so there exists a constant
c=c(ξ) >0suchthat|ξ−p/q|>c/q^2 for every rational numberp/q. It is a long-
standing conjecture that this is false for any real algebraic numberξof degreed>2.
It is not difficult to show that Roth’s theorem may be restated in the following
homogeneous form: if
L 1 (u,v)=αu+βv, L 2 (u,v)=γu+δv,are linearly independent linear forms with algebraic coefficientsα,β,γ,δ, then, for
any givenε>0, there exist at most finitely many integersx,y, not both zero, such that
|L 1 (x,y)L 2 (x,y)|<max(|x|,|y|)−ε.Thesubspace theoremof W. Schmidt (1972) generalizes Roth’s theorem in this
form to higher dimensions. In the stronger form given it by Vojta (1989) it says:
if L 1 (u),...,Ln(u) are linearly independent linear forms in n variables
u = (u 1 ,...,un)with (real or complex) algebraic coefficients, then there exist
finitely many proper linear subspacesV 1 ,...,Vh ofQn such that every nonzero
x=(x 1 ,...,xn)∈Znfor which
|L 1 (x)···Ln(x)|<‖x‖−ε,