Higher diophantine approximation exponents and continued fraction symmetries for function fields

Abstract:

We construct many families of nonquadratic algebraic Laurent series with continued fractions having a bounded partial quotients sequence (the diophantine approximation exponent for approximation by rationals is thus $2$, agreeing with the Roth value) and with the diophantine approximation exponent for approximation by quadratics being arbitrarily large. In contrast, the Schmidt value (analog of the Roth value for approximations by quadratics, in the number field case) is $3$. We calculate diophantine approximation exponents for approximations by rationals for function field analogs of $\pi$, $e$ and Hurwitz numbers (which are transcendental) and also give an interesting lower bound (which may be the actual value) for the exponent for approximation by quadratics for the latter two. We do this exploiting the situation when ‘folding’ or ‘negative reversal’ patterns of the relevant continued fractions become ‘repeating’ or ‘half-repeating’ in even or odd characteristic respectively.