Diophantine Approximation and Integral Points on Curves
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.
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