Algebraic numbers are not near many rationals
In mathematics, Roth's theorem or Thue–Siegel–Roth theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational approximations that are 'very good'. Over half a century, the meaning of "very good" here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955).
Statement.
Roth's theorem states that every irrational algebraic number formula_1 has approximation exponent equal to 2. This means that, for every formula_2, the inequality
formula_3
can have only finitely many solutions in coprime integers formula_4 and formula_5. Roth's proof of this fact resolved a conjecture by Siegel. It follows that every irrational algebraic number α satisfies
formula_6
with formula_7 a positive number depending only on formula_2 and formula_1.
Discussion.
The first result in this direction is Liouville's theorem on approximation of algebraic numbers, which gives an approximation exponent of "d" for an algebraic number α of degree "d" ≥ 2. This is already enough to demonstrate the existence of transcendental numbers. Thue realised that an exponent less than "d" would have applications to the solution of Diophantine equations and in Thue's theorem from 1909 established an exponent formula_10 which he applied to prove the finiteness of the solutions of Thue equation. Siegel's theorem improves this to an exponent about 2√"d", and Dyson's theorem of 1947 has exponent about √2"d".
Roth's result with exponent 2 is in some sense the best possible, because this statement would fail on setting formula_11: by Dirichlet's theorem on diophantine approximation there are infinitely many solutions in this case. However, there is a stronger conjecture of Serge Lang that
formula_12
can have only finitely many solutions in integers "p" and "q". If one lets α run over the whole of the set of real numbers, not just the algebraic reals, then both Roth's conclusion and Lang's hold
for almost all formula_1. So both the theorem and the conjecture assert that a certain countable set misses a certain set of measure zero.
The theorem is not currently effective: that is, there is no bound known on the possible values of "p","q" given formula_1. showed that Roth's techniques could be used to give an effective bound for the number of "p"/"q" satisfying the inequality, using a "gap" principle. The fact that we do not actually know "C"(ε) means that the project of solving the equation, or bounding the size of the solutions, is out of reach.
Proof technique.
The proof technique involves constructing an auxiliary multivariate polynomial in an arbitrarily large number of variables depending upon formula_15, leading to a contradiction in the presence of too many good approximations. More specifically, one finds a certain number of rational approximations to the irrational algebraic number in question, and then applies the function over each of these simultaneously (i.e. each of these rational numbers serve as the input to a unique variable in the expression defining our function). By its nature, it was ineffective (see effective results in number theory); this is of particular interest since a major application of this type of result is to bound the number of solutions of some Diophantine equations.
Generalizations.
There is a higher-dimensional version, Schmidt's subspace theorem, of the basic result. There are also numerous extensions, for example using the "p"-adic metric, based on the Roth method.
William J. LeVeque generalized the result by showing that a similar bound holds when the approximating numbers are taken from a fixed algebraic number field. Define the "height" "H"(ξ) of an algebraic number ξ to be the maximum of the absolute values of the coefficients of its minimal polynomial. Fix κ ;2. For a given algebraic number α and algebraic number field "K", the equation
formula_16
has only finitely many solutions in elements ξ of "K".
Notes.
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