On the Distribution of the Digits in Luroth Expansions

For x is an element of [0, 1), let [d(1)(x), d(2)(x), ...] be its Luroth expansion, and let {p(n)(x)/q(n)(x)}(n >= 1) be the sequence of convergents of x. In this paper, we prove that the Hausdorff dimension of the exceptional set F-alpha(beta) = {x is an element of [0, 1): lim inf(n ->infinity) log d(n+1)(x)/-log vertical bar x - p(n)(x)/q(n)(x)vertical bar = alpha, lim sup(n ->infinity) log d(n+1)(x)/-log vertical bar x - p(n)(x)/q(n)(x)vertical bar >= beta} is (1 - beta)/2 or 1 - beta according to alpha > 0 or alpha = 0. This extends an earlier result of Tan and Zhang.