Hausdorff dimensions of some irregular sets associated with β-expansions

The Hausdorff dimensions of some refined irregular sets associated with beta-expansions are determined for any beta > 1. More precisely, Hausdorff dimensions of the sets {x is an element of [0,1) : lim inf(n ->infinity) S-n(x,beta)/n = alpha(1), lim sup(n ->infinity) S-n(x,beta)/n = alpha(2)}, alpha(1), alpha(2) >= 0 are obtained completely, where S (n) (x, beta) = I pound (k = 1) (n) epsilon (k) (x, beta) denotes the sum of the first n digits of the beta-expansion of x. As an application, we present another concise proof of that the set of points x a [0, 1) satisfying does not exist is of full Hausdorff dimension.