Thue–Siegel–Roth theorem
Encyclopedia
In mathematics
, the Thue–Siegel–Roth theorem, also known simply as Roth's theorem, is a foundational result in diophantine approximation
to algebraic number
s. It is of a qualitative type, stating that a given algebraic number α 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 Liouville in 1844 and continuing with work of , , , and .
can have only finitely many solutions in integers p and q, as was conjectured by Siegel. Therefore any irrational algebraic α satisfies
with C(α,ε) a positive number depending only on ε > 0 and α.
Roth's result is in some sense the best possible, because this statement would fail on setting ε = 0: by Dirichlet's theorem on diophantine approximation there are infinitely many solutions in this case. For comparison, Liouville's theorem has exponent about d, Thue's theorem from 1909 has exponent
, Siegel's theorem has exponent about 2√d, and Dyson's theorem has exponent about √(2d) where d ≥ 2 is the degree of α. 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
α. So both the theorem and the conjecture assert that a certain countable set misses a certain set of measure zero.
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 bounding the number of solutions of some diophantine equation
s.
showed that Roth's techniques could be used to give an effective bound for the number of rational approximations to algebraic numbers, but not for their sizes. 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. Later work using the methods of Alan Baker made some small impact on effective improvements to Liouville's theorem on diophantine approximation, which gives a bound
(see Liouville number); but the inequalities are still weak.
generalized the result by showing that a similar bound holds when the approximating numbers are taken from a fixed algebraic number field
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Thue–Siegel–Roth theorem, also known simply as Roth's theorem, is a foundational result in diophantine approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
to algebraic number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
s. It is of a qualitative type, stating that a given algebraic number α may not have too many rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a 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 Liouville in 1844 and continuing with work of , , , and .
Statement
The Thue–Siegel–Roth theorem states that any irrational algebraic number α has approximation exponent equal to 2, i.e., for given ε > 0, the inequalitycan have only finitely many solutions in integers p and q, as was conjectured by Siegel. Therefore any irrational algebraic α satisfies
with C(α,ε) a positive number depending only on ε > 0 and α.
Roth's result is in some sense the best possible, because this statement would fail on setting ε = 0: by Dirichlet's theorem on diophantine approximation there are infinitely many solutions in this case. For comparison, Liouville's theorem has exponent about d, Thue's theorem from 1909 has exponent
, Siegel's theorem has exponent about 2√d, and Dyson's theorem has exponent about √(2d) where d ≥ 2 is the degree of α. However, there is a stronger conjecture of Serge LangSerge Lang
Serge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra...
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
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...
α. So both the theorem and the conjecture assert that a certain countable set misses a certain set of measure zero.
Proof technique
The proof technique was the construction of an auxiliary functionAuxiliary function
In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high...
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
Effective results in number theory
For historical reasons and in order to have application to the solution of Diophantine equations, results in number theory have been scrutinised more than in other branches of mathematics to see if their content is effectively computable...
); this is of particular interest since a major application of this type of result is to bounding the number of solutions of some diophantine equation
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
s.
showed that Roth's techniques could be used to give an effective bound for the number of rational approximations to algebraic numbers, but not for their sizes. 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. Later work using the methods of Alan Baker made some small impact on effective improvements to Liouville's theorem on diophantine approximation, which gives a bound
(see Liouville number); but the inequalities are still weak.
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, based on the Roth method. LeVequeWilliam J. LeVeque
William Judson LeVeque was an American mathematician and administrator who worked primarily in number theory. He was executive director of the American Mathematical Society during the 1970s and 1980s when that organization was growing rapidly and greatly increasing its use of computers in academic...
generalized the result by showing that a similar bound holds when the approximating numbers are taken from a fixed algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
.