Markov and Lagrange Spectra for Laurent series in 1/T with rational coefficients

Abstract

The field of formal Laurent series is a natural analogue of the real numbers, and mathematicians have been translating well-known results about rational approximations to that setting. In the framework of power series over the rational numbers, we define and study the Lagrange spectrum, related to Diophantine approximation of irrationals, and the Markov spectrum, related to representation by indefinite binary quadratic forms. We compute both spectra explicitly, and show that they coincide and exhibit no gaps, contrary to what happens over the reals.