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Concept
Roth's theorem
Summary
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 number 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 , , , and .
Roth's theorem states that every irrational algebraic number has approximation exponent equal to 2. This means that, for every , the inequality
can have only finitely many solutions in coprime integers and . Roth's proof of this fact resolved a conjecture by Siegel. It follows that every irrational algebraic number α satisfies
with a positive number depending only on and .
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 which he applied to prove the finiteness of the solutions of Thue equation. Siegel's theorem improves this to an exponent about 2, and Dyson's theorem of 1947 has exponent about .
Roth's result with exponent 2 is in some sense the best possible, because this statement would fail on setting : 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
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 . So both the theorem and the conjecture assert that a certain countable set misses a certain set of measure zero.
Official source
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