Diophantine approximation

The mathematical discipline of Diophantine approximation, named after Diophantos of Alexandria, originally dealt with the approximation of real numbers by rational numbers. Well-known theorems in the theory of Diophantine approximation are the Dirichlet approximation theorem and the Thue-Siegel-Roth theorem. More generally, the field can be defined as the approximation of zero by real functions with finitely many integer arguments.

The theory also plays a significant role in the question of the solvability of Diophantine equations and in the theory of transcendental numbers. Diophantine inequalities are often considered.

Euler proved in the 18th century that the best rational approximations of real numbers are given by the approximate fractions of their regular continued fraction expansion (if you break off the continued fraction at a point, you have a rational number as an approximation to the real number). That {\tfrac {p}{q}}is a best approximation of xmeans that

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

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

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

{\displaystyle \left|qx-p\right|<\left|q^{\prime }x-p^{\prime }\right|}

Best approximations in the sense of this second definition are also best approximations in the sense of the first definition, but not vice versa. For regular continued fractions, the n-th approximations are best approximations in the sense of the second definition (see continued fraction and other results given there).

Joseph Liouville proved in 1844 that for algebraic numbers (solutions of an algebraic equation of degree nwith integer coefficients) there is a lower bound for the approximation by rational numbers that depends on the denominator of the rational number and on the degree of the equation:

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

with a constant cdepends only on the number to be approximated. The theorem can be interpreted in such a way that irrational algebraic numbers cannot be approximated "very well" by rational numbers. Liouville thus also achieved the first proof of the existence of a transcendental number, because if one finds an irrational number that can be approximated "very well" by rational numbers (i.e. better than is possible by the restrictions of Liouville's theorem), it cannot be algebraic (Liouville numbers). Liouville's theorem was tightened over time until the Thue-Siegel-Roth theorem in the 20th century with an exponent {\displaystyle 2+\varepsilon }in the denominator at the lower bound and a constant that additionally depended on the arbitrarily small real number ε \varepsilon

An upper bound for the approximation by rational numbers is given by Dirichlet's approximation theorem: For each real number xthere are infinitely many rational approximations {\tfrac {p}{q}}with

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

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


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