| 摘 要： | The Hausdorff dimensions of some refined irregular sets associated with β-expansions are determined for any β 1. More precisely, Hausdorff dimensions of the sets {x ∈ 0, 1) :lim inf(n→∞) S_n(x, β)/n= α_1, lim sup (n→∞) S_n(x, β)/n= α_2}, α_1, α_2≥0 are obtained completely, where S_n(x, β) =sum ε_k(x, β) from k=1 to n denotes the sum of the first n digits of the β-expansion of x. As an application, we present another concise proof of that the set of points x ∈ 0, 1) satisfying lim_(n→∞) S_n(x,β)/n does not exist is of full Hausdorff dimension.
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