Metric theorems for continued β-fractions

• Published in: Monatshefte für Mathematik Vol. 190; no. 2; pp. 281 - 299
• Main Authors: Feng, Jing, Ma, Chao, Wang, Shuailing
• Format: Journal Article
• Language: English
• Published: Vienna Springer Vienna 01.10.2019; Springer Nature B.V
• Subjects: Divergence; Mathematics; Mathematics and Statistics; Polynomials; Quotients; Integers; Partial quotients; 28A80; 11K55; Continued; fractions
• ISSN: 0026-9255; 1436-5081
• DOI: 10.1007/s00605-019-01305-6

Let β > 1 be a root of the polynomial t 2 = a t + 1 with a ∈ N , a ≥ 1 or a root of the polynomial t 2 = a t - 1 with a ∈ N , a ≥ 3 . In this paper, we consider the metric properties of the continued β -fractions. We show that the Lebesgue measure of the following set E ( φ ) = { x ∈ [ 0 , 1 ) : a n ( x ) ≥ φ ( n ) for infinitely many n ∈ N } is null or full according to the convergence or divergence of the series ∑ n = 1 ∞ 1 φ ( n ) , where a n ( x ) is the n -th partial quotients in the continued β -fraction expansion of x and φ is a postive function defined on N . As a result, the set of numbers in the interval [0, 1) with bounded partial quotients in their continued β -fraction expansions is of zero Lebesgue measure.