Diophantine approximation

The mathematical discipline of Diophantine approximation, named after Diophantos of Alexandria , originally deals with the approximation of real numbers by rational numbers . Well-known sentences in the theory of Diophantine approximation are the Dirichlet approximation theorem and the Thue-Siegel-Roth theorem . More generally, the area can be defined as an approximation of zero by real functions with a finite number of integer arguments.

Theory also plays an important role in the question of the solvability of Diophantine equations and in the theory of transcendent numbers . Diophantine inequalities are often considered.

Euler proved in the 18th century that the best rational approximations of real numbers by the regular continued fractions are given (breaks to the continued fraction at one point down, one has a rational number as an approximation to the real number). The fact that is a best approximation of means that ${\ displaystyle {\ tfrac {p} {q}}}$ ${\ displaystyle x}$

{\ displaystyle \ left | x - {\ frac {p} {q}} \ right | \ leq \ left | x - {\ frac {p '} {q'}} \ right |}

for every rational number with - that every better approximation has a larger denominator. ${\ displaystyle {\ tfrac {p '} {q'}}}$ ${\ displaystyle 0 <q '\ leq q}$

Sometimes the following inequality is also used to define the best approximation:

{\ displaystyle \ left | q \, xp \ right | <\ left | q ^ {\ prime} \, xp ^ {\ prime} \ right |}

Best approximations in the sense of this definition are also best approximations in the sense of the first definition, but not the other way around. With regular continued fractions are th convergents best approximations in the sense of the second definition (see continued fraction and other specified there results). ${\ displaystyle n}$

Joseph Liouville proved in 1844 that for algebraic numbers (solutions of an algebraic equation of degree with integer coefficients) there is a lower bound for the approximation by rational numbers, which depends on the denominator of the rational number and on the degree of the equation: ${\ displaystyle n}$

{\ displaystyle \ left | x - {\ frac {p} {q}} \ right |> {\ frac {c} {q ^ {n}}}}

with a constant dependent on the number to be approximated . The theorem can be interpreted in such a way that irrational algebraic numbers of degree greater than 1 are difficult to approximate by rational numbers. Liouville also succeeded in proving the existence of a transcendent number for the first time, because one finds an irrational number that can be approximated better by rational numbers than in Liouville's theorem, it cannot be algebraic ( Liouville numbers ). Liouville's theorem was tightened over time up to Thue-Siegel-Roth's theorem in the 20th century with an exponent in the denominator at the lower bound and a constant that also depended on the arbitrarily small real number . ${\ displaystyle c}$ ${\ displaystyle 2+ \ epsilon}$ ${\ displaystyle \ epsilon}$

Dirichlet's approximation theorem gives an upper bound for the approximation by rational numbers : For every irrational algebraic number there is an infinite number of rational approximations with ${\ displaystyle x}$ ${\ displaystyle {\ frac {p} {q}}}$

{\ displaystyle \ left | x - {\ frac {p} {q}} \ right | <{\ frac {1} {q ^ {2}}}.}

On the right-hand side, the denominator can still be improved to ( Émile Borel ), further tightening is not possible according to Hurwitz's theorem , since there are only finitely many solutions for approximating the golden number for in the denominator with . ${\ displaystyle {\ sqrt {5}} \, q ^ {2}}$ ${\ displaystyle c \, q ^ {2}}$ ${\ displaystyle c> {\ sqrt {5}}}$

literature

JF Koksma : Diophantine Approximations. Results of mathematics and its border areas, Springer, Berlin 1936.
JWS Cassels : An introduction to diophantine approximation. Cambridge Tracts in Mathematics and Mathematical Physics 45, Cambridge University Press, 1957.
Vladimir G. Sprindžuk : Metric theory of Diophantine approximations . John Wiley & Sons, New York NY et al., 1979, ISBN 0-470-26706-2 .
Wolfgang M. Schmidt : Diophantine approximation. Lecture Notes in Mathematics 785, Springer, 1980.
Serge Lang : Introduction to Diophantine Approximations , New Expanded Edition. Edition, Springer-Verlag, New York NY et al., 1995, ISBN 0-387-94456-7 .

Web links

Sprindzhuk, Diophantine Approximations, Encyclopedia of Mathematics , Springer

References and comments

↑ Wladimir Gennadjewitsch Sprindschuk , Diophantine Approximation, Encyclopedia of Mathematics, Springer, see web links.
↑ Feldman, Algebraic and transcendental numbers, Quantum, July / August 2000, p. 23. The sentence is used that if an algebraic number of degree n is greater than 1 root of a polynomial with integer coefficients, this polynomial has no rational roots .

[1] Wladimir Gennadjewitsch Sprindschuk , Diophantine Approximation, Encyclopedia of Mathematics, Springer, see web links.

[2] Feldman, Algebraic and transcendental numbers, Quantum, July / August 2000, p. 23. The sentence is used that if an algebraic number of degree n is greater than 1 root of a polynomial with integer coefficients, this polynomial has no rational roots .