# Roth's theorem

(Redirected from Thue-Siegel-Roth theorem)

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 ${\displaystyle \alpha }$ 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

Roth's theorem states that every irrational algebraic number ${\displaystyle \alpha }$ has approximation exponent equal to 2, i.e., for given ${\displaystyle \varepsilon >0}$, the inequality

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2+\varepsilon }}}}$

can have only finitely many solutions in coprime integers ${\displaystyle p}$ and ${\displaystyle q}$, as was conjectured by Siegel. Therefore every irrational algebraic α satisfies

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|>{\frac {C(\alpha ,\varepsilon )}{q^{2+\varepsilon }}}}$

with ${\displaystyle C(\alpha ,\varepsilon )}$ a positive number depending only on ${\displaystyle \varepsilon >0}$ and ${\displaystyle \alpha }$.

## 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 ${\displaystyle d/2+1+\varepsilon }$. Siegel's theorem improves this to an exponent about 2d, 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 ${\displaystyle \varepsilon =0}$: 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

${\displaystyle \left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{2}\log(q)^{1+\varepsilon }}}}$

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 ${\displaystyle \alpha }$. 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 ${\displaystyle \alpha }$. [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

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

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

${\displaystyle |\alpha -\xi |<{\frac {1}{H(\xi )^{\kappa }}}}$

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