论文标题
Cantor系列中的描述性复杂性
Descriptive complexity in Cantor series
论文作者
论文摘要
相对于基本序列$ q =（q_1，q_2，\ dots）$的cantor系列扩展，其中$ q_i \ geq 2 $是表格$ x = a_0 + \ sum_ = a_0 + \ sum_ {i = 1}^\ infty \ infty \ frac \ frac \ frac \ frac {a _i} $ quots的$ x = a_0 + \ sum_} a_i <q_i $。这些概括了普通的基础$ b $扩展，其中$ q_i = b $。 Ki和Linton表明，对于普通基础$ b $扩展，正常数的集合是$ \boldsymbolπ^0_3 $ -Complete Set，确定了该集合的确切复杂性。在康托尔系列的情况下，有三个自然概念：正态性，比率正态性和分布正常（这些概念对于基本$ b $扩展是等效的）。我们表明，对于任何$ q $，套装$ \ mathscr {dn}（q）$ fordriase number是$ \boldsymbolπ^0_3 $ -complete，如果$ q $是$ 1 $ -DDIVERGENT（即$ \ sum_，$ \ sum_ = 1} $ \ MATHSCR {n}（q）$和$ \ MATHSCR {rn}（q）$正常数为$ \boldsymbolπ^0_3 $ -complete。我们进一步表明，这些集合的所有五个非平地差异均为$ d_2（\boldsymbolπ^0_3）$ - 如果$ \ \ lim_i q_i = \ infty $和$ q $是$ 1 $ -Divergent（$ \ sedscr {n}（q）\ setMinus \ nerts $ nimsscr \ n.这表明，除了限制$ \ mathscr {n}（q）\ subseteq \ mathscr {rn}（q）$之外，这三个概念尽可能独立。
A Cantor series expansion for a real number $x$ with respect to a basic sequence $Q=(q_1,q_2,\dots)$, where $q_i \geq 2$, is a representation of the form $x=a_0 + \sum_{i=1}^\infty \frac{a_i}{q_1q_2\cdots q_i}$ where $0 \leq a_i<q_i$. These generalize ordinary base $b$ expansions where $q_i=b$. Ki and Linton showed that for ordinary base $b$ expansions the set of normal numbers is a $\boldsymbolΠ^0_3$-complete set, establishing the exact complexity of this set. In the case of Cantor series there are three natural notions of normality: normality, ratio normality,and distribution normality (these notions are equivalent for base $b$ expansions). We show that for any $Q$ the set $\mathscr{DN}(Q)$ of distribution normal number is $\boldsymbolΠ^0_3$-complete, and if $Q$ is $1$-divergent (i.e., $\sum_{i=1}^\infty \frac{1}{q_i}$ diverges) then the sets $\mathscr{N}(Q)$ and $\mathscr{RN}(Q)$ of normal and ratio normal numbers are $\boldsymbolΠ^0_3$-complete. We further show that all five non-trivial differences of these sets are $D_2(\boldsymbolΠ^0_3)$-complete if $\lim_i q_i=\infty$ and $Q$ is $1$-divergent (the trivial case is $\mathscr{N}(Q)\setminus \mathscr{RN}(Q)=\emptyset$). This shows that except for the containment $\mathscr{N}(Q)\subseteq \mathscr{RN}(Q)$, these three notions are as independent as possible.
