EDB — 0C1

Exercises

1. [0C1]Prerequisites:[0BW]↺↻. (Dirichlet’s approximation theorem) Given an irrational number $x$, show that there are infinitely many rationals $𝛼$ such that we can represent $𝛼=m/n$ in order to satisfy the relation

   $$|x-\frac{m}{n}|<\frac{1}{n^2}.$$

   Some comments.

   • Note for every fixed $n\ge 2$ there is at most an $m$ for which the previous relation holds; but there may not be one.

   • Note that if the relation holds for a rational $𝛼$, there are only finite choices of representations for which it holds,

   • and certainly it holds for the "canonical" representation with $n,m$ coprimes.

   Solution 1

   [0C2]↺↻

   [2B0]↺↻