Diophantine Approximation and Integral Points on Curves

Abstract

The fundamental problem in the subject of Diophantine approximation is the question of how closely an irrational number can be approximated by a rational number. For example, if α ∈ ℝ is any given real number, we may ask how closely can one approximate α by a rational number p/q ∈ ℚ The obvious answer is that the difference (p/q) — α∣ can be made as small as desired by an appropriate choice of p/q. This is nothing more than the assertion that ℚ is dense in ℝ. The problem is to show that if the difference is small, then p and q must be large.

He was a poet and hated the approximate. R. M. Rilke, Tie Journal of My Other Self