Diophantine Approximation and its Applications

Abstract

As with many other areas, it is difficult to say when the development of the theory of diophantine approximation started. Diophantine equations have been solved long before Diophantos of Alexandria (perhaps A.D. 250) wrote his books on Arithmetics. Diophantos devised elegant methods for constructing one solution to an explicitly given equation, but he does not use inequalities. Archimedes’s inequalities 3 10/71 < π < 3 1/7 and Tsu Ch’ung-Chih’s (A.D. 430–501) estimate 355/113 = 3.1415929... for π = 3.1415926... are without any doubt early diophantine approximation results, but the theory of continued fractions does not have its roots in the construction methods for finding good rational approximations to π. but rather in the algorithm developed by Brahmagupta (A.D. 628) and others for finding iteratively the solutions of the Pell equation x² − dy² = 1. Euler proved in 1737 that the continued fraction expansion of any quadratic irrational number is periodic. The converse was proved by Lagrange in 1770. Lagrange deduced various inequalities on the convergents of irrational real numbers. In particular, he showed that every irrational real α admits infinitely many rationals p/q such that

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

(1.1)