Approximation lattices of p-adic numbers

Approximation lattices occur in a natural way in the study of rational approximations to p-adic numbers. Periodicity of a sequence of approximation lattices is shown to occur for rational and quadratic p-adic numbers, and for those only, thus establishing a p-adic analogue of Lagrange's theorem on periodic continued fractions. Using approximation lattices we derive upper and lower bounds for the best approximations to a p-adic number, thus establishing the p-adic analogue of a theorem of Hurwitz.