LECTURE 12: DIOPHANTINE APPROXIMATION 1. DIRICHLET THEOREM Many important ideas in Number Theory stem from notions of Diophantine approximation, which is to say rational approximations to real numbers with prescribed properties. Theorem 1.1 (Dirichlet). Let 0 E R and let Q be a real number exceeding 1. Then there exist integers p and q with 1 < q < Q and (p, q) = 1 such that q6 - pl ? 1/Q. Proof. We apply the Box Principle. Write N = [Q], and consider the N + 1 real numbers 0, 1, {0}, {20}, ., {(N - 1)0}, where here, and throughout, we write {x} for x - [x]. These N + 1 real numbers all lie in the interval [0, 1]. But if we divide this unit interval into N disjoint intervals of length 1/N, it follows that there must be two numbers from the above set which necessarily lie in the same interval. The difference between these two numbers has the shape q0 - p, where p and q are integers with 0 < q < N. Thus we deduce that there exist integers p and q with 1 ? q < Q and |q0 - p| ? 1/Q. The coprimality condition is obtained easily by dividing through by (p, q). Corollary 1.2. Whenever 0 is irrational, there exist infinitely many distinct pairs p E Z and q E N with (p, q) = 1 and |0 - p/q| < 1/q2. Proof. Let Q > 1. Then by Dirichlet's theorem on Diophantine approximation, there exist p E Z and q E N with (p, q) = 1, q < Q and 0 < |0 - p/q| < 1/(qQ) < 1/q2. Let Q' be any real number exceeding |0 - p/q|-1. A second application of Dirichlet's theorem shows that there exist p' E Z and q' E N with (p', q') = 1, 1 < q < Q' and ? p 0 6 q'Q' 1 10 - p/q| p Thus, necessarily, one has p/q + p/q. Furthermore, 90 - p /q'| < 1/(q)2. By iterating this process, we obtain a sequence (Pn/In) n=1 of distinct rational numbers with . Pn 10 Pn-1 P1 , 0 < < <.... and |0 - pi/qi| < 1/q", and hence infinitely many approximations p/q with (p,q) = 1 and |0 - p/q| <1/q2. 1
2 LECTURE 12 2. CONTINUED FRACTIONS Given a rational fraction w with uo E Z and u1 E N, we apply the Euclid algorithm to obtain Wo = aou1 + U2, 0 < U2 < < 1, U1= @12 + U3, 0 < 3 < U2, 0 < Un+1 < Un, . Un-1 = an-1un + Un+1, Un = anun+1 If we set 0; = ui, then we obtain the relation 0; = ai + 01+1 1 , i=0 ... ,n - 1, 0n = an. This gives the expansion u0 =00=ao+ a1 + U1 1 1 . 1 @2 + 1 . . . + an For the