Ergodic properties of some inverse polynomial series expansions of laurent series

Abstract

Recently A. Knopfmacher and the present author [8] introduced and studied some properties of various unique expansions of formal Laurent series over a field F, as the sums of reciprocals of polynomials, involving "digits" al, as , . . , lying in a polynomial ring F[X] over F. In particular, one of these expansions (described below) turned out to be analogous to the so-called Liiroth expansion of a real number, discussed in Perron [13] Chapter 4. In a partly parallel way, Artin [1] and Magnus [9, 10] had earlier studied a Laurent series analogue of simple continued fractions of real numbers, involving "digits" Xl,Z2,. . . in a polynomial ring as above. In addition to sketching elementary properties of an n-dimensional "Jacobi-Perron" variant of this, Paysant-Leroux and Dubois [11, 12] also briefly outlined certain "metric" theorems analogous to some of Khintchine [7] for real continued fractions, in the case when F is a finite field. The main aim of this paper is to derive some similar metric or ergodic results for the Laurent series Liirothtype expansion referred to above. (For analogous results concerning Liiroth expansions of real numbers, see :lager and de Vroedt [5] and Salgt [14], and also [16, 17, 18].) In order to explain the conclusions, we first fix some notation and describe the inverse-polynomial Liiroth-type representation to be considered: