Beta-expansion and continued fraction expansion over formal Laurent series

摘要

Let x ∈ I be an irrational element and n ≥ 1, where I is the unit disc in the field of formal Laurent series F((X~(-1))), we denote by k_n(x) the number of exact partial quotients in continued fraction expansion of x, given by the first n digits in the β-expansion of x, both expansions are based on F((X~(-1))). We obtain that lim inf_(n→+∞)(k_n(x)) = (deg β)/(2Q~*(x)), lim sup_(n→+∞)(k_n(x)) = (deg β)/(2Q_*(x)), where Q~*(x), Q_*(x) are the upper and lower constants of x, respectively. Also, a central limit theorem and an iterated logarithm law for {k_n(x)}_n ≥ 1 are established.