Beta-expansion and continued fraction expansion over formal Laurent series

Let x∈Ix∈I be an irrational element and n⩾1n⩾1, where I   is the unit disc in the field of formal Laurent series F((X−1))F((X−1)), we denote by kn(x)kn(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))F((X−1)). We obtain thatlim infn→+∞kn(x)n=degβ2Q*(x),lim supn→+∞kn(x)n=degβ2Q*(x), where Q*(x),Q*(x)Q*(x),Q*(x) are the upper and lower constants of x  , respectively. Also, a central limit theorem and an iterated logarithm law for {kn(x)}n⩾1{kn(x)}n⩾1 are established.