Introduction to Diophantine Approximation. Part II

Introduction to Diophantine Approximation

In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ¬ 1/x, where θ is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. F...

Roth’s Theorem: an introduction to diophantine approximation

Indeed, P (p/q) is a sum of rational numbers whose denominators are all factors of q: expressing this as a rational number with denominator q, it is either identically zero or it is at is at least 1/q in absolute value because one is the smallest positive integer. Since |P (p/q)| is bounded below as a function of q, when it is non–zero, it follows from a continuity argument that p/q can not be ...

Introduction to Part Ii

In the following chapters we present the theory of bifurcations of dynamical systems with simple dynamics. It is difficult to over-emphasize the role of bifurcation theory in nonlinear dynamics the reason is quite simple: the methods of the theory of bifurcations comprise a working tool kit for the study of dynamical models. Besides, bifurcation theory provides a universal language to communica...

Introduction to Part II

Diophantine Approximation of Ternary Linear Forms . II

Let 6 denote the positive root of the equation xs + x2 — 2x — 1 = 0; that is, 8 = 2 cos(27r/7). The main result of the paper is the evaluation of the constant lim supm-co min M2\x + By + 02z|, where the min is taken over all integers x, y, z satisfying 1 g max (\y\, |z|) g M. Its value is (29 + 3),/7 = .78485. The same method can be applied to other constants of the same type.