Some exceptional sets of Borel-Bernstein theorem in continued fractions

Let [a(1)(x), a(2)( x), a(3)(x), ... ] denote the continued fraction expansion of a real number x is an element of [0, 1). This paper is concerned with certain exceptional sets of theBorel-Bernstein Theorem on the growth rate of {a(n)(x)}(n >= 1). As a main result, the Hausdorff dimension of the set E-sup(psi) = {x is an element of [0, 1) : lim sup(n ->infinity) log a(n)(x)/psi(n) = 1} is determined, where psi : N -> R+ tends to infinity as n ->infinity.