Abstract :
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.
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