中文摘要：

给定一个正整数序列Q={q_k}_k≥1,其中q_k≥2.任意的x∈[0,1]对应唯一的Q-Cantor展式.令T_Qn（x）=q_1···q_nx-「q_1···q_nx」.对于任意的正函数φ：N→（0,1）和序列y={y_n}≥1？[0,1],本文考虑集合E_y（φ）：={x∈[0,1]：｜T_Qn（x）-y_n｜〈φ（n）i.o.n}的大小,指出了集合E_y（φ）的Lebesgue测度和Hausdorff测度结果只依赖特定级数的敛散性,与y={y_n}_（n≥1）无关.

英文摘要：

Let Q = {q_k}_（k≥1） be a sequence of positive integers with q_k ≥ 2 for every k ≥ 1. Then each point x ∈ [0, 1] is attached with an infinite series expansion which is called the Q-Cantor series expansion of x. Put T_Qn（x） = q_1 · · · q_nx-「q_1· · · q_nx」. For any positive function φ ： N →（0, 1） with φ（n） → 0 as n → ∞ and any sequence y = {y_n}_（n≥1） [0, 1],we consider the size of the set E_y（φ） ：= {x ∈ [0, 1] ： ｜T_Qn（x）-y_n｜ φ（n） i. o. n}. In this paper, we show that both the Lebesgue measure and the Hausdorff measure of E_y（φ） fulfill a dichotomy law according to the divergence or convergence of certain series.