Beta-expansion and continued fraction expansion ✩

For any real number β >1, let ε(1,β) = (ε1(1), ε2(1), . . . , εn(1), . . .) be the infinite β-expansion of 1. Define ln = sup{k 0: εn+j (1) = 0 for all 1 j k}. Let x ∈ [0, 1) be an irrational number. We denote by kn(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n digits in the β-expansion of x. If {ln, n 1} is bounded, we obtain that for all x ∈ [0, 1) \Q, lim inf n→+∞ kn(x) n = logβ 2β∗(x) , lim sup n→+∞ kn(x) n = logβ 2β∗(x) , where β∗(x), β∗(x) are the upper and lower Lévy constants, which generalize the result in [J. Wu, Continued fraction and decimal expansions of an irrational number, Adv. Math. 206 (2) (2006) 684–694].Moreover, if lim supn→+∞ ln n = 0, we also get the similar result except a small set. © 2007 Elsevier Inc. All rights reserved.