Roth's theorem

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In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that a given algebraic number may not have too 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 Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955).

Statement[edit]

Roth's theorem states that every irrational algebraic number has approximation exponent equal to 2, i.e., for given , the inequality

can have only finitely many solutions in coprime integers and , as was conjectured by Siegel. Therefore every irrational algebraic α satisfies

with a positive number depending only on and .

Discussion[edit]

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 . Siegel's theorem improves this to an exponent about 2√d, and Dyson's theorem of 1947 has exponent about √(2d).

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.[1]

The theorem is not currently effective: that is, there is no bound known on the possible values of p,q given . [2] Davenport & Roth (1955) 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.[2] 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[edit]

The proof technique was the construction of an auxiliary function in several variables, leading to a contradiction in the presence of too many good approximations. 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[edit]

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,[3] based on the Roth method.

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

has only finitely many solutions in elements ξ of K.[4]

See also[edit]

Notes[edit]

  1. ^ It is also closely related to the Manin–Mumford conjecture.
  2. ^ a b Hindry, Marc; Silverman, Joseph H. (2000). Diophantine Geometry: An Introduction. Graduate Texts in Mathematics. 201. pp. 344–345. ISBN 0-387-98981-1. 
  3. ^ Ridout, D. (1958). "The p-adic generalization of the Thue–Siegel–Roth theorem". Mathematika. 5: 40–48. Zbl 0085.03501. doi:10.1112/s0025579300001339. 
  4. ^ LeVeque, William J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. pp. II:148–152. ISBN 978-0-486-42539-9. Zbl 1009.11001. 

References[edit]

Further reading[edit]