Markov and Lagrange spectra for Laurent series in $1/T$ with rational coefficients
Volume 200 / 2021
Acta Arithmetica 200 (2021), 17-38
MSC: Primary 11J06; Secondary 11E16.
DOI: 10.4064/aa200325-27-1
Published online: 7 June 2021
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 Markov spectrum, related to representation by indefinite binary quadratic forms, and the Lagrange spectrum, related to Diophantine approximation of irrationals. We compute both spectra explicitly, and show that they coincide and exhibit no gaps, in contrast to what happens over the reals.
