A fundamental result in Diophantine approximation stating that any irrational algebraic number α cannot be approximated by rationals p/q arbitrarily closely: for every ε>0, the inequality |α−p/q|<1/q^{2+ε} has only finitely many rational solutions p/q. Equivalently, the approximation exponent of an algebraic irrational is 2.

Principle

Algebraic irrationals resist rational approximation beyond the quadratic Dirichlet threshold; the theorem reduces the possible approximation exponent to its minimal Diophantine value 2, up to arbitrarily small error factors.

Demonstration

Concrete instance: for α=√2 and any fixed ε>0 there are only finitely many rationals p/q satisfying |√2−p/q|<1/q^{2+ε}, unlike Liouville-type numbers which admit infinitely many extremely good approximations.

Misapplication

Treating Roth's theorem as effective by asserting explicit bounds or algorithms for all sufficiently good approximations; Roth's method is ineffective and does not provide a computable bound for when the finite set is exhausted.

Consequence

Leads to finiteness results for many Diophantine inequalities and underpins later refinements and transference results; it distinguishes algebraic irrationals sharply from transcendental numbers in approximation quality.

Reversal

By contrast, transcendental numbers (and Liouville numbers in particular) can be approximated arbitrarily well by rationals, exhibiting approximation exponents greater than 2 or even infinite exponent, so the Roth conclusion fails for them.

Boundary

Applies only to irrational algebraic numbers; it does not apply to rational numbers (degree 1) nor directly to transcendental numbers, and it yields finiteness but not effective lists of exceptions.

Semantic Tension

Tension exists between Roth's qualitative finiteness and effective results: some theorems in Diophantine approximation give explicit constants (e.g., Baker-type results) while Roth gives the optimal exponent but no effective constants, which can be confused.

Synthesis

Roth's Theorem asserts that algebraic irrationals have Diophantine approximation exponent exactly 2: they admit at most finitely many rational approximations exceeding the 1/q^{2+ε} threshold, a decisive qualitative barrier separating algebraic from many transcendental approximation behaviors.