Beta-expansion and continued fraction expansion over formal Laurent series

Abstract

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 ) n = deg β 2 Q * ( x ) , lim sup n → + ∞ k n ( x ) n = deg β 2 Q * ( 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.