Metric theorems for continued β-fractions

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.