SWhat changed
Some irrational numbers can be approximated extraordinarily well by fractions. Roth proves a sharp limit for real algebraic irrationals: for any fixed positive epsilon, only finitely many reduced fractions p/q satisfy |alpha−p/q|<q^(−2−epsilon). At exponent two, every irrational still has infinitely many good approximations. His proof closes the rational-approximation exponent gap left by earlier work without supplying a general computable list of all exceptional fractions.
Scientific scope
Some irrational numbers can be approximated extraordinarily well by fractions. Roth proves a sharp limit for real algebraic irrationals: for any fixed positive epsilon, only finitely many reduced fractions p/q satisfy |alpha−p/q|<q^(−2−epsilon). At exponent two, every irrational still has infinitely many good approximations. His proof closes the rational-approximation exponent gap left by earlier work without supplying a general computable list of all exceptional fractions. This is a finiteness theorem for approximation of a fixed algebraic irrational by rationals. It is not a uniform computable denominator cutoff, an effective algorithm listing all exceptional approximants, a bound for arbitrary transcendental targets, a statement that the exponent 2 inequality has only finitely many solutions, or the distinct Roth arithmetic-progression theorem. The distinct generalization on page 2 to approximants of arbitrary algebraic degree is outside this rational-approximation record.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. This is a finiteness theorem for approximation of a fixed algebraic irrational by rationals. It is not a uniform computable denominator cutoff, an effective algorithm listing all exceptional approximants, a bound for arbitrary transcendental targets, a statement that the exponent 2 inequality has only finitely many solutions, or the distinct Roth arithmetic-progression theorem. The distinct generalization on page 2 to approximants of arbitrary algebraic degree is outside this rational-approximation record.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
- H. Davenport · SCIENCE: Encouragement, manuscript rewriting and proof simplifications, including the replacement by Lemma 8, acknowledged by Roth; not a target-paper byline author.
Provisional WRS assessment, edition 2026-10-23.4. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
Full scientific record · Browse contribution grades · Request a correction