chapter 2

38 Pages

## Continued fractions

The digit expansion of an irrational real number z produces rational approximations with arbitrary exactitude. If we use (say) g-adic digit expansions, we usually need a dominator of size N = gn to accomplish an approximation with an error less than N−1 = g−n. In contrast, Dirichlet’s approximation theorem shows that a suitable choice of denominator q gives rise to an approximation with an error less than q−2 (see Theorem 1.2.3). A systematic way to produce such good approximations with small denominators is provided by the theory of continued fractions, which is a main tool in the theory of Diophantine approximation.