Diophantine Approximation in Finite Characteristic

In contrast to Roth’s theorem that all algebraic irrational real numbers have approximation exponent two, the distribution of the exponents for the function field counterparts is not even conjecturally understood. We describe some recent progress made on this issue. An explicit continued fraction is not known even for a single non-quadratic algebraic real number. We provide many families of explicit continued fractions, equations and exponents for non-quadratic algebraic laurent series in finite characteristic, including non-Riccati examples with both bounded or unbounded sequences of partial quotients. On this occasion of Professor Abhyankar’s 70th birthday conference, it might be appropriate to mention some recent applications of the ‘high school algebra’ [A] to the study of diophantine approximation for function fields in finite characteristic. This study is related to some of his loves: power series, continued fractions, algebraic curves, finite characteristic, resultants (and even automata). 1 What we know and don’t know about the basic questions The term ‘irrational’ suggests a need to ‘rationalize’ and one of the basic questions of diophantine approximation is how well we can approximate irrational real numbers by rationals. Since the rationals are dense in reals, we can make error arbitrarily small, so the question really is how small we can make it relative to the complexity (height) of the rational approximation measured traditionally by the size of its denominator. Let us recall some basic history. The details and references can be found in the papers in the bibliography, for example, in [S1] for the number field case, and [S2], [T] for function field case. A simple application of the Dirichlet box principle applied to the fractional parts of multiples of α and to the boxes consisting of equal sized sub-intervals of the interval (0, 1) (or approximation by convergents of continued fraction) shows that given irrational α, there are infinitely many rationals p/q satisfying |α− p/q| < 1/q. On the other hand, if the irrational α is algebraic of degree d = deg(α), then (as Liouville showed) applying the mean value theorem ? * Supported in part by NSA and NSF grants