On continued fraction expansions of quadratic irrationals in positive characteristic

Frederic Paulin, Uri Shapira

Research output: Contribution to journalArticlepeer-review

Abstract

Let R D Fq[Y ] be the ring of polynomials over a finite field Fq, let K D Fq((Y -1) be the field of formal Laurent series over Fq, let f 2 K be a quadratic irrational over Fq.Y / and let P 2 R be an irreducible polynomial. We study the asymptotic properties of the degrees of the coefficients of the continued fraction expansion of quadratic irrationals such as P n f as n→+ ∞, proving, in sharp contrast with the case of quadratic irrationals in R over Q considered in [1], that they have one such degree very large with respect to the other ones. We use arguments of [2] giving a relationship with the discrete geodesic flow on the Bruhat–Tits building of .PGL2; K/ and, with A the diagonal subgroup of PGL2.K/ y , the escape of mass phenomena of [7] for A-invariant probability measures on the compact A-orbits along Hecke rays in the moduli space PGL2.R/n PGL2.K/ y

Original languageEnglish
Pages (from-to)81-105
Number of pages25
JournalGroups, Geometry, and Dynamics
Volume14
Issue number1
DOIs
StatePublished - 2020

Keywords

  • Artin map
  • Bruhat-Tits tree
  • Bruhat–Tits tree
  • Continued fraction expansion
  • Hecke tree
  • Positive characteristic
  • Quadratic irrational
  • continued fraction expansion
  • positive characteristic

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Geometry and Topology

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