Beta-expansion and continued fraction expansion over formal Laurent series
The authors study the relationship between the analogues of continued fractions and \(\beta\)-expansions in fields of formal Laurent series with coefficients from a finite field \(\mathbb F((X^{-1}))\). Their main result generalizes an earlier result by the authors [J. Math. Anal. Appl. 339, No. 2, 1322--1331 (2008; Zbl 1137.11053)]. Briefly, let \(\beta \in \mathbb F((X^{-1}))\) with \(\|\beta \| > 1\), where \(\| \cdot \|\) denotes the usual absolute value on \(\mathbb F((X^{-1}))\). Write any \(x\) in the unit ball of \(\mathbb F((X^{-1}))\) as a \(\beta\)-expansion as well as a continued fraction, \[ x = \sum_{i=1}^\infty {{\varepsilon_i(x)}\over{\beta^i}} = {1 \over {A_1(x) + {1 \over{A_2(x) + \cdots}}}}. \] Let \(J(\varepsilon_1, \dots, \varepsilon_n)\) denote the set of power series which have \(\varepsilon_1, \dots, \varepsilon_n\) as the first \(n\) digits in their \(\beta\)-expansion, and let \(I(A_1, \dots, A_n)\) denote the set of power series with \(A_1, \dots, A_n\) as the first \(N\) partial qoutients in their continued fraction expansion. To relate these two expansions, let \[ k_n(x) = \max\{m \geq 0 : J(\varepsilon_1(x), \dots, \varepsilon_n(x)) \subseteq I(A_1(x), \dots, A_m(x))\}. \] The authors study the growth and distribution of the sequence \(k_n(x)\). It is shown that if \(x \in \mathbb F((X^{-1}))\) is irrational, \textit{i.e.}, \(x\) is not the ratio of two polynomials in \(\mathbb F[X]\), then \[ \liminf_{n \rightarrow \infty} {{k_n(x)}\over n} = {{\deg\beta} \over {2 Q^*(x)}} \quad \text{and}\quad \limsup_{n \rightarrow \infty} {{k_n(x)}\over n} = {{\deg\beta} \over {2 Q_*(x)}}, \] where \[ Q^*(x) = \limsup_{n \rightarrow \infty} {{\deg(Q_n(x))}\over n} \quad \text{and}\quad Q_*(x) = \liminf_{n \rightarrow \infty} {{\deg(Q_n(x))}\over n}, \] with \(Q_n(x)\) denoting the denominator of the \(n\)'th convergent of \(x\) from the continued fraction expansion. As a consequence, a Central Limit Theorem and a Law of the Iterated Logarithm for \(k_n(x)\) are derived.
scientific article; zbMATH DE number 5316918
scientific article