Approximation Exponents for Function Fields

For diophantine approximation of algebraic real numbers by rationals, Roth’s celebrated theorem settles the issue of the approximation exponent completely in the number field case. But in spite of strong analogies between number fields and function fields, its naive analogue fails in function fields of finite characteristic, and the situation is not even conjecturally understood. In this short survey, we describe the recent progress on this and the related issues of explicit continued fractions for algebraic quantities. 1. Exponents of Diophantine Approximation How well we can approximate an irrational real number α by rationals a/b, compared to their complexity, traditionally measured by the size of b, is a central question of diophantine approximation theory. Thus we define the approximation exponent of α by E(α) := lim sup ( − log |α− a/b| log |b| )