Modular group, continued fractions and Diophantine approximation in characteristic \(p\) (Q1863079)

Continued fraction expansions of formal Laurent series in \(x^{-1}\) over a finite field were already used by E. Artin in his Ph.D. dissertation, published in 1924; the role of the ``integer'' part here is played by the ``polynomial'' part. The Diophantine and metric properties of these continued fractions have recently been studied, see say \textit{W. M. Schmidt} [Acta Arith. 95, No. 2, 139-166 (2000; Zbl 0987.11041)], and \textit{V. Berthé} and \textit{H. Nakada} [Expo. Math. 18, No. 4, 257-284 (2000; Zbl 1024.11050)]. Here, the author gives a clear geometric interpretation of this continued fraction expansion in terms of geodesics of the Bruhat-Tits tree for the special linear group of the Laurent series field. Results include geometric proofs of (quantitative versions of) the facts: convergents give best approximations; quadratics have periodic expansions; and, elements algebraic over the Laurent series field are poorly approximated.